I ! !l : U II I tii ■ Hi / *\/" '/ Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/electricalphenom01pernuoft \* SEMICENTENNIAL PUBLICATIONS OF THE UNIVERSITY OF CALIFORNIA 1868-1918 ELECTRICAL PHENOMENA IN PARALLEL CONDUCTORS VOLUME I ELEMENTS OF TRANSMISSION BY FREDERICK EUGENE PERNOT, Ph.D. Assoc. Member A. I.E. E.; Assistant Professor of Electrical Engineering, University of California; Captain, Signal Reserve Corps, U.S.A. FIRST EDITION Ifefffl^O IS', 19 21 NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited 1918 Copyright, 1918, BY FREDERICK EUGENE PERNOT Stanhope £CC3M F. H. GILSON COMPANT BOSTON, U.S.A. PREFACE It is of fundamental importance to be able to predetermine with as much numerical accuracy as possible, the phenomena which may take place in electrical systems. It is of importance first, because such predetermination, when possible, can be made at much less expense and trouble than is required to obtain the same information experimentally; and second, because information so obtained Opens a way to new developments and improvements in the application of scientific knowledge. To predetermine in a numerical way what may happen in electrical systems under different conditions requires that the various phenomena be expressed in a mathematical form or the equivalent thereof. Of course it is not the mathematical expressions themselves which are of main interest, but these expressions are necessary since they afford the only means at our disposal, other than experimental investigation, by which the desired knowledge may be obtained. In this volume an attempt has been made to set forth the mathematical developments leading to solutions for a number of problems arising in connection with the transmission of electrical energy over metallic circuits. The fundamental equations for the propagation of currents and voltages along an electrical circuit consisting of parallel conductors or the equivalent thereof have been developed and presented by a number of writers — Heaviside, Fleming, Pupin, Steinmetz, Kennelly and others; but it has been the writer's experience to note that the application of these de- velopments to present-day engineering problems has been com- paratively rare. This has been due partly to the unfamiliarity of a great many of engineers with the particular type of mathe- matics necessarily involved in such discussions, and partly to the fact that the presentation of the material in various publications has not been in such a form as particularly to invite its application. Instead of developing and applying to problems rigorous methods of solution, the tendency has been to develop apparently simple approximate methods; but investigation shows that in many cases IV PREFACE the simplicity is only apparent and that accurate results may be obtained with as little mechanical labor as the approximate. Further, a realization or understanding of accurate methods leads to an ability to develop approximate methods where such are needed on account of the complexity of rigorous solutions, and to apply these approximate methods with due consideration of their limitations and significance. For some time the author has felt that it would be of real servifie to have collected under one head as much as possible of the available material dealing with the present subject, as well aa the addition of such further developments as may be possible. It seemed desirable, also, to include in this work the discussion of a number of matters, which, though not falling directly under the above title, are nevertheless used so often in connection with the particular problems which do fall under this head that a complete understanding thereof is essential. Accordingly, in this volume a discussion of periodic and alternating quantities in general, with methods of analysis of periodic curves into Fourier's series, has been included. A discussion of the more common form of oscillograph, methods for its use and an analysis of its mode of operation, was deemed desirable. The fact that at the present date the oscillo- graph plays such an important part in the study of transmission- line phenomena leads to the belief that its discussion here is not irrelevant. In regard to the treatment of transmission-line phenomena proper, this volume is to be thought of as forming an introduction to subsequent volumes dealing with specialized forms of electrical transmission, and therefore it naturally has for its scope a dis- cussion of some of the more general properties of transmission systems only. Continuously alternating-current phenomena only have been discussed, for the introduction of methods for deter- mining the transient currents or voltages in a conductor or system of conductors leads to solutions more complicated than seem advisable here. Further, desirable forms of such solutions differ so greatly for different types of circuits which may arise that no one particular form could be considered sufficiently general to be treated in this introductory volume. Between points of discontinuity in a simple line carrying con- tinuously alternating electrical quantities, the currents and volt- ages at one point are related to the currents and voltages at PREFACE V another point by simple linear relations, and these relations are naturally expressed in terms of hyperbolic functions. Since in alternating-current work complex quantities are used for the representation, analytically, of the vectors in a vector diagram, our algebraic formulas necessarily involve hyperbolic functions of complex variables. This fact has been one of the greatest obstacles preventing a more general utilization of rigorous trans- mission-line formulas. The portion of this volume which deals with transmission-line phenomena may properly be thought of as merely an elaboration dealing with the various forms and results which may be obtained from the two fundamental linear equations relating quantities at one point in a line to similar quantities at another point. In many places these linear equations are of exactly the same form as the equations for systems of entirely different types, — that is, localized circuits, — and therefore our discussions, instead of applying to transmission lines only, are applicable also to the general electrical circuit for which equations of the same nature hold true. The coefficients of such linear equations are of course given by different functional relations among the various circuit constants when different types of circuits are considered. In " Theory and Calculation of Electric Currents " by J. L. LaCour and 0. J. Bragstad a discussion of the general electrical circuit is given, with reference to the operating charac- teristics of circuits carrying uniformly alternating electrical quan- tities, which the author has found very illuminating. It is to be hoped that further treatments from this standpoint may appear, for the generality of the methods employed make such discussions of great value. Although, in this present volume, some explanation of the complex-quantity method of dealing with alternating quantities seemed desirable in order to establish for the reader a clear knowl- edge of the forms of procedure followed, it has necessarily been assumed that those interested would be familiar with the various fundamental theorems and ideas concerning the behavior of the simple properties which go to make up an electrical circuit. Hav- ing clearly in mind the physical significance of the four funda- mental constants involved in electrical systems, that is, resistance, conductance, self-inductance and electrostatic capacity, and hav- ing formulated the methods by which mathematical discussions relating to alternating quantities may be carried on, further VI PREFACE developments may be thought of as only mathematical in their nature. The above condition, however, does not obviate the necessity of being very familiar, from a physical standpoint, with the particular problems in hand, for unless such familiarity exists much difficulty will be encountered in carrying through the mathe- matical developments leading to solutions which are of interest and importance to the practicing engineer and to the physicist. The author wishes to express here his appreciation of the valuable advice and encouragement offered by Professor Harris J. Ryan of Stanford University and of the services rendered by Messrs. E. N. D'Oyly and Geo. L. Greves in verifying the various mathematical steps and numerical illustrations. FREDERICK EUGENE PERNOT. Berkeley, California. April 27, 1918. CONTENTS Section Page Preface iii CHAPTER I Phenomena in Continuous Current Transmission over Non-Leaky Lines 1. The Simple Direct Current Circuit 1 2. Non-uniform Conductors 1 3. Transmission Phenomena 3 CHAPTER II Direct Current Line with Leakage 4. General 7 5. Fundamental Differential Equation and Solution 7 6. Determination of Integration Constants 8 7. Solution in Terms of Hyperbolic Functions 9 8. Particular Solutions for g = and r = 10 9. Load-end Quantities in Terms of Generator-end Quantities 11 10. Voltage Regulation 12 11. Power Relations in General 13 12. Maximum Power 14 13. Efficiency 15 14. Numerical Illustration 17 15. Effective Resistance of Line and Determination of Constants by Measurement 20 16. Combinations of Leaky Lines 24 17. Single Generator and Line Supplying Two Loads 25 18. Single Load Supplied from Two Power Sources 26 19. Unloaded Line with Double Source of Power Supply 28 20. Numerical Illustration 29 21. General Networks « 30 CHAPTER III Periodic and Alternating Quantities. Fourier's Series and Analysis of Periodic Curves 22. General 33 23. Periodic Quantities and Fourier's Theorem 33 24. Use of Fourier's Series 33 vii viii CONTENTS Section Page 25. Determination of Unknown Coefficients by Integration 34 26. Use of the Auxiliary Curves, y' = y cos nx and y" = y sin nx 35 27. Determination of Coefficients from a Finite Number of Ordinates . . 35 28. The Cosine Terms 35 29. The Sine Terms 37 30. Interpretation of Results — Cosine Terms 38 31. Interpretation of Results — Sine Terms 39 32. Particular Orders of Harmonics 39 33. Limitations to the Use of a Finite Number of Ordinates 40 34. Summary 41 35. Mechanical Analysis based upon Integration Processes 42 36. Mathematical Theory of Analyzer 43 37. Tables and Forms for Analysis from a Number of Ordinates 45 38. Selection of the Number of Ordinates. Separation of Harmonics . . 45 39. Description of Tabular Forms for Complete Analysis 46 40. Summary 48 41. Numerical Illustration and Check of Tabular Forms 49 42. Comparative Accuracy 53 CHAPTER IV Treatment op Non-sinusoidal Alternating Quantities. The Use of the Oscillograph 43. General 60 44. The Polar Diagram and Representation of Alternating Quantities . . 61 45. Complex Quantity Representation 63 46. The Complex Operator — Ratio between Two Alternating Quanti- ties Expressed as Vectors , 65 47. Transformation from Trigonometric Functions of Time to Complex Quantity Representation 67 48. Superposition. Use and Limitations of the Complex Quantity Method 67 49. Numerical Example of Complex Quantity Treatment of Non- sinusoidal Alternating Quantities 69 50. Distorting Circuits for Use with the Oscillograph 73 51. Oscillographic Determination of Wave Shape of E.M.F 76 CHAPTER V Equations of Motion of the Galvanometer Mirror in the Electromagnetic Type of Oscillograph 52. General , 81 53. Fundamental Equations of Motion of the Vibrator Strips 82 54. Determination of Integration Constants — Their Dependence upon the Moment of Inertia of the Mirror 83 55. The Mirror Motion 85 56. Application of Analysis to a Particular Case 86 CONTENTS k Section Page 57. The Intensity of the Magnetic Field Surrounding the Strips 90 58. Electromotive Force Induced in the Vibrator Strips 91 59. The Free Vibration Frequency of the System when the Inertia of the Mirror is Considered 91 CHAPTER VI Alternating Current Phenomena in an Ideal Line 60. The Ideal Line 95 61. Fundamental Circuit Constants 95 62. Fundamental Differential Equations and Solution 96 63. Auxiliary Circuit Constants 97 64. Complete Expansion of Exact Solution 98 65. Interpretation of the Various Terms in Expanded Solution 98 66. Example of Calculation of the Fundamental Quantities in the Expanded Form of Equation 100 67. Vector Diagrams Representing the Exponential Form of Equations 102 68. Operating Characteristics from Vector Diagrams 107 69. Limiting Forms of Vector Diagrams for Infinite Length Line 108 70. Instantaneous Values from Vector Diagrams 108 71. Instantaneous Values of Power at Different Points '. Ill 72. Equations for the Loci of the Ends of the Component Vectors .... Ill 73. Mechanical Construction of the Equiangular Spirals 113 74. Introduction of Hyperbolic Functions of Complex Variables 114 75. Calculation of Constants in Equations for a Line of Fixed Length. . 115 76. Graphical Treatment for Various Load-end Conditions 117 77. Power-factor from Vector Diagrams 120 78. Operating Characteristics of Line with Various Loads 120 79. Constant Generator Voltage — Rigorous Solution 125 80. Constant Generator Voltage — Solution by Plotted Curves 127 81. Infinite Series Form for Transmission Line Solution 133 82. Illustration of Convergence of Series 134 83. Line Admittance and Impedance Localized 135 84. Collection of Formulae for Use in Computing. Tabular Arrangement 138 CHAPTER VII The Propagation Constant V = VZY = a +j0 85. General 141 86. Fundamental Equations for a and 141 87. Wave Length and Velocity of Propagation 142 88. Values of a, /3, and S when gL = rC 143 89. Infinite Frequency Values of a, 0, and S 144 90. Low Frequency Values of a, /3, and S 145 91. Illustration of Variation of a, /3, and *S, with Frequency, for Power Circuit 146 92. Relations between a, 0, and S i 149 93. Numerical Illustrations for Telephone Circuit 150 X CONTENTS CHAPTER VIII The Equivalent Impedance of Circuits and the Determination of the Fundamental Constants by Measurement Section Paob 94. General Expression for Generator-end Impedance 153 95. Generator-end Impedances under Special Conditions 154 96. Analogies between Circuits with Distributed and with Lumped Constants 156 97. Limiting Conditions at Infinite Frequencies 160 98. Infinite-line Impedance 161 99. Numerical Illustrations of Equivalent Line Impedance 162 100. Determination of Line Constants from Short-circuit and Open- circuit Tests 165 101. Multiple Solutions 169 102. Methods of Making Impedance Measurements 169 103. An Impedance Bridge 170 104. Precautions in the Use of the Bridge 171 105. Illustrative Computation 173 106. Effect of Errors in Original Data 173 CHAPTER IX Change of Wave Shape Along Lines 107. Method of Treatment of Complex Wave Shapes 177 108. General Case — Load-end Impedance = Z 177 109. Short-circuited and Open-circuited Lines 178 110. Numerical Illustration. 100 Mile Open-circuited Line 181 111. Equations for Instantaneous Voltage 183 112. Equations for Instantaneous Currents 184 113. Discussions of Wave Shapes 185 114. Dependence of Wave Shapes upon Generator and Load Character- istics 188 115. Double Source of Energy Supply through Terminal Impedances . . 189 116. Approximations Based upon the Neglect of Line Losses 191 117. Numerical Comparison of Approximate and Exact Solutions 193 118. Resonance Frequencies 194 CHAPTER X Power Relations in Long Lines 119. Instantaneous and Average Power 198 120. The Separate Components of E and / at any Point 198 121. The Power Equation 199 122. Reduction in Form of the Power Equation 200 123. Final Form of the Power Equation 202 124. Numerical Example 202 125. Line Losses 204 CONTENTS Xi Section Page 126. Unloaded Line 204 127. Power in Distortionless Lines 204 128. Load Impedance for a Minimum Line Loss with a given Po 207 129. Numerical Illustration 209 130. Line Efficiency 210 131. Load Power-factor for Maximum Efficiency 211 132. Numerical Illustration. Variation of Efficiency with Power- factor 213 133. Peculiarities Incident to Maximum-efficiency Transmission 216 134. Variation of Maximum-efficiency Conditions with Line Length . . . 217 135. Zero Length Lines 218 136. Infinite Length Lines 220 137. General 221 138. Proof that for Maximum Efficiency the Ratio between e and eo Equals the Ratio between i and io, and that the Generator Power-factor Angle is Equal and Opposite to the Load Power- factor Angle 222 139. Load Voltage for Maximum All-day Efficiency with Intermittent Loads 223 140. Line Loss with Varying Loads, and the Proper Load Voltage to Secure a Minimum of Such Loss 225 141. Illustrative Example from Daily Load Curve 227 142. Dependence of Proper Voltage upon the Leakage Coefficient 228 143. Determination of Resistance and Leakage Losses Separately; Graphical Method 229 144. Illustration of Graphical Method 231 145. Separate Losses Analytically 232 146. Illustration of Analytical Method 235 147. Separate Losses at No Load 235 CHAPTER XI Voltage Regulation Case I. Both Components of the Load Become Zero Simultaneously 148. Preliminary Developments 237 149. Scalar Values 238 150. Introduction of General Form of Equations 238 151. Reactive Volt-amperes Required at Load for Zero Regulation. . . . 239 152. Conditions for Minimum Voltage Regulation 240 153. Conditions for any Voltage Regulation 241 154. Solution in Terms of Load Admittance 242 155. Numerical Illustration 242 156. Nature of Curves Relating g and bo 243 157. Geometrical Construction of Curves 245 158. Relations between Vector Diagrams and Foregoing Curves 246 159. General 248 XU CONTENTS Case II. Total Load-end Susceptance Constant Section Pagb 160. Load-end Voltages in Terms of Load-end Admittances 249 161. Constant Susceptance Required for Given Regulation 250 162. Real and Imaginary Solutions 250 163. Numerical Illustration 251 164. Nature of the Curves 252 165. Significance of the Coordinates of the Central Point 254 166. Mechanical Construction of Hyperbolas 254 Case III. A Portion Only of the Load-end Susceptance Constant 167. Load-end Voltages 257 168. Constant Portion of Load-end Susceptance for a Given Regulation . 258 169. Nature and Location of Curves 259 170. Mechanical Construction of the Curves 259 171. Forms of Curves in Special Cases 261 172. Zero Voltage Regulation 262 Case IV. Load of Constant Power-factor 173. Equation Relating Constant Susceptance and Regulation 264 174. Location of Curves 264 175. Nature of Curves 265 176. Mechanical Construction 266 177. Numerical Illustration 267 Appendix Hyperbolic Functions 269 Relationships Involving Hyperbolic and Allied Functions 272 Tables of Logarithms of Hyperbolic Functions 285 Tabular Forms for Use in the Analysis of Periodic Curves 325 ELECTEICAL PHENOMENA IN PARALLEL CONDUCTORS CHAPTER I PHENOMENA IN CONTINUOUS CURRENT TRANSMIS- SION OVER NON-LEAKY LINES 1. The Simple Direct Current Circuit. — The simplest con- ception of a transmission circuit is that in which an unvarying current is .transmitted over a system in which only the ohmic resistance of the conductors need be considered. By taking into account only the resistance of the conductors, the tacit assumption is made that between any two points which constitute the ends of the transmission system under discussion (either as actual termini or as points at which discontinuities are introduced) there is no "leakage" of current from one conductor or "side" of the system to the other. The electric current is therefore the same in magni- tude at both ends of such a system, while the potential difference, or voltage, between the conductors changes from point to point because of the electromotive forces consumed by the resistance and current flow. 2. Non-uniform Conductors. — The general case for the above is that in which the resistance of the conductors per unit length along the system is not constant. Physically, such- a condition brings to mind a circuit in which the conductors are of varying cross-section or material. Since the cross-section or material of the conductor is varying from point to point, the resistance of the conductor per unit length along the circuit must be defined by the equation dR /1 x r= dT (1) 2 CONTINUOUS CURRENT TRANSMISSION in which dR is the actual resistance of the circuit included within the length dl. This expression can be made to include both wires of the system if they are parallel and therefore of equal lengths. Let A m cross-sectional area of conductor. p «■ specific resistance of conductor material. Then dR = dR\ -+* dRt — — j — I — 2 — » in which the subscripts 1 and 2 refer to the separate wires or conductors. Unless otherwise specified, r will be understood to be defined by (2), thereby including both conductors. Since r varies along the line, in order that the problem be determinate its value must be given as a function of distance, from one end of the line say, and therefore r=fd). Obviously, from (2) the total line resistance of the system is JrVo Pk rdl= I f(l)dl (3) «/0 If E = voltage or potential difference between the conductors at any point, the gradient of potential difference at this point is defined by dE F = -jj- = gradient of potential difference. (4) If / = current in the conductors and the distance I be counted positively from the load end towards the source of energy, F = ™ = rl = hf(l). (5) The integral of (5) gives the expression for line voltage at any point, E = Jl f(l)dl = IoR + C. R = line resistance between the point in question and the load, and therefore the integration constant C is seen to be equal to the voltage at the load, E . E = E Q + Rh, (6) TRANSMISSION PHENOMENA 3 and at the supply end, where I = lo, E = E + R Jo. (7) In short, it is seen that even though the resistance be distributed in a non-uniform manner, it is entirely unnecessary to consider anything except the total value of line resistance, Rt= J rdl, unless the manner of distribution of the various electrical quanti- ties throughout the line length is desired. This simplicity arises from the fact that the line current is uniform or constant in value throughout the line length, and therefore all integrals reduce to expressions involving only the total resistance. Though very simple, this matter is one of fundamental importance, for it is due to departures from this condition that the more complicated solutions become necessary. 3. Transmission Phenomena. — Utilizing the value of total line resistance R t , the entire discussion of such a transmission line is nothing more than that of a simple electrical circuit. Eq = voltage at load or receiving end. 7o = current throughout the system. Rt = total line resistance. Then E = E + R Jo = generator voltage. (8) Po = Eolo = power at load, in watts. (9) P = EIo = Po + Rth 2 = power at generator. (10) P' = RJo 2 = total line loss. (11) P E Eff. ■■ -=• «» -=• ■« efficiency of transmission. (12) Mr £j The voltage regulation of a transmission line is defined as rise in receiving end voltage from load to no load load voltage the generator voltage being supposed to remain constant after throwing off the load. Under load conditions, by equation (8) the receiving end voltage is Eq = E — RtIo> 4 CONTINUOUS CURRENT TRANSMISSION and at no load, therefore, Eq = E. The rise in voltage from load to no load is RJ . R I Reg- = p — f ~B~T = regulation at load 7 . (13) h — tiuo As will be noted, equation (13) is not that of a straight line with 7 as independent variable if the generator voltage E be supposed constant for all loads. If, however, the generator voltage be adjusted with changing load so that the receiver voltage E remains constant, then the regulation will obviously be given by- Reg. = -4-^, for constant E . With a constant generator voltage E, it is interesting to notice the change in the power at the receiver, P , with a change in cur- rent I . p = EJo = (E- RJ ) h = Eh - RJo 2 . (14) From this equation it is seen that P is a quadratic function of I , and therefore for any particular value of P there are two possible values of current I . Also, for certain values of P the expressions for Jo become complex imaginaries, indicating physically that the value of P for which the solution was made was larger than the line in question is able to transmit at the assumed valjue of genera- tor voltage. This is, of course, for positive values of P . The maximum amount of power which may be transmitted over the line of resistance R t , at a generator voltage E, is to be obtained from (14). By differentiation thus ^ = E - 2 RJo = 0, for -maximum P , (15) alo E 7o = 17-5- , for maximum P , (16) Z lit and the maximum power becomes, by substituting (16) in (14), (Po)max. = ~ (16a) For the current value determined by (16) it is obvious that the load resistance is equal to the line resistance, R = Rt, which is the TRANSMISSION PHENOMENA i.W and E i Volu 130 260 120 240 %/ XW\X 110 220 100 200 vE„ 90 180 p / 80 160 BH. L ae/ Los l ro 140 00 120 50 1D0 40 80 80 60 P« 20 40 10 20 00 120 180 240 420 540 Fig. 1. 300 360 I Amperes Direct Current Transmission Line Characteristics. Line Resistance, 0.50 ohm. Generator Voltage Constant, 250 Volts. 1 k.wJ and] % Volts 120 240 \ E H 110 220 Kej 100 200 80 180 _£ff. 80 160 V 70 140 60 120 50 100 40 80 BO 60 ^ 20 40 10 20 R phn 8 tm i l'>ss 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Fig. 2. — Direct Current Transmission Line Characteristics. Line Resistance, 0.50 ohm. Generator Voltage Constant, 250 Volts. 6 CONTINUOUS CURRENT TRANSMISSION familiar relation between internal or line resistance, and load resistance, for a maximum power at the receiver. Although in the preceding all quantities have been expressed in terms of the load current and load voltage, it may be convenient to introduce as independent variable the load resistance R . Thus, substituting E = R h in equation (12) for line efficiency, and simplifying the expression, Ko T" tit which is an expression independent of either the voltage or current. Similarly, in equation (13) for line regulation, Reg. - f. (18) For power, since E T — inrl T? — T? T A o — r, i D ana ejq — nolo, Ko "T txt E* P = EIo = r= — r— ^ = power at generator. Ho -r ttt (19) T> Po = Roh 2 = /p , d \ 2 E* = power at load. (20) For the case of constant generator voltage, Fig. 1 shows a number of curves plotted to 7 as independent variable, and Fig. 2 shows the curves for the same system plotted with the load resist- ance R as independent variable. The above discussion covers the more important phenomena in a simple direct-current transmission where line resistance only need be considered. The equations are intended to cover only the case of a line supplied with power at one end and loaded at the other. Obviously, in the -practical distribution of power the simple con- ditions mentioned are not often realized. Such problems as the design of a railway feeder to supply power to an interurban rail- way, for instance, involve much more than a consideration of the simple matters mentioned above. There will be no attempt made here to take up such matters as distribution problems, particu- larly as there are already a number of publications dealing exclu- sively with the practice and economics of power distribution. CHAPTER II DIRECT-CURRENT LINE WITH LEAKAGE 4. General. — Under this head will be discussed the case of a line whose conductors are of uniform resistance and between which there exists a uniformly distributed conductance, or means of leakage. A portion of the following has already been presented in numerous works, but it is included here for the sake of com- pleteness. The following discussion is based entirely upon the use of hyper- bolic functions, the natural means of expression for the phenomena occurring in systems of the character named. In fact, throughout the entire discussion of transmission systems, the use of hyperbolic functions leads to an elegance of treatment which cannot be at- tained by any other means. 5. Fundamental Differential Equation and Solution. — Let r = resistance of both conductors per unit length of line, ohms. g = conductance from wire to wire per unit length of line, mhos. I = distance counted positively towards energy supply. E = line voltage at any point at distance, I. I = line current at any point at distance, I. The resistance of an element of line of length dl is r dl, and the conductance between wires is g dl. Therefore, taking differentials dI = Egdl, dE = Irdl, dl „ dE T Ji = E9 > Ti =Ir - (i) Differentiating again with respect to I, (2) <H_dE dP ~ g dl' cPE = dl 7 dl 2 r dl 8 DIRECT-CURRENT LINE WITH LEAKAGE Substituting the values of the first derivatives from (1) in (2), ** T O w -rgE = 0. (3) The two equations in (3) are identical in form, and are of the general type of linear, second order differential equation of which the general integral is y = CtP" + C# m *, in which mi and m^ are the two roots of the quadratic own? + aim -f a* = 0, and C\ and C2 are integration constants. In (3), the particular case under discussion, ao = 1, ai = 0, and 02 = — rg, which gives m? — rg = 0, mi = -r-Vrg, (4) mi = —y/rg. The solutions of (3) are therefore E = A^ 1 + igr^ (5) where Ai, A 2 , Bi, and B 2 are integration constants to be deter- mined by initial conditions. 6. Determination of Integration Constants. — As initial con- ditions, let the current and voltage at the load, or receiving end, be designated by 7 and E , respectively. At this end, by the arbitrary choice of positive direction along the line 1 = and the exponential terms in (5), reduce to unity. To evaluate the integration constants, differentiate the first of SOLUTIONS IN TERMS OF HYPERBOLIC FUNCTIONS 9 (5) and substitute the result in the second equation of (1) for the point I = 0. Thus, dE dl. Also, by (5), E 1=0 Ji=o = VrgAi — VrgA 2 = rl = A x + A 2 = E Q . (6) The solution of the two simultaneous equations (6) yields (7) By symmetry, or by proceeding in an exactly similar manner, 2 \ * r / V?). and therefore, in the general integral (5), i-J[(*+ji\^.^+(*-i.^)-^ (8) (9) Equation (9) is complete, in that it expresses the voltage and current at any point explicitly in terms of the line constants and the voltage and current at the load end. In general, it will be found that it is most convenient to so utilize the load quantities as independent variables, though for some purposes other pro- cedures may be more desirable. This matter will be discussed later, in connection with the performance of lines carrying alter- nating currents. 7. Solutions in Terms of Hyperbolic Functions. — Although complete, equation (9) is not in very convenient form for compu- tation. By combining terms, and letting v = Vrg, (10) 10 DIRECT-CURRENT LINE WITH LEAKAGE a better form results. *-|(*«+ «-«)*■ (ID The parentheses are immediately seen to constitute the two hyper- bolic functions, cosh vl and sinh vl, which will be utilized therefor. The equations then become E = E cosh vl + 7 y - sinh t>Z, 7 = Jo cosh «2 + #o y - sinh aZ. (12) v = Vrg may be considered as one of the fundamental line constants, as well as the quantity y - or its reciprocal y -. In fact, these two quantities completely specify the characteristics of a direct-current transmission circuit when operating under steady conditions, and as far as computation purposes are con- cerned they constitute a more convenient means of specification than the simple quantities r and g, though either set is easily obtained from the other. For future convenience in writing, the line constants will be expressed in either one of the alternative forms, as follows: r = line resistance per unit length, g = leakage conductance per unit length, or v = *JTg, and * = y^, y = \ = \/l- (13) The fundamental line equations, on which subsequent develop- ments are based, are then E = E cosh vl + JoZ sinh vl, I = To cosh vl -+- E y sinh vl. (14) 8. Particular Forms of Solutions f or g = and for r = 0. If either r or g becomes zero, then equations (12) or (14) assume an indeterminate form, which may be evaluated either by the standard LOAD-END QUANTITIES 11 methods of calculus for the evaluation of indeterminate forms, or by a consideration of the actual physical phenomena occasioned by one of these constants becoming equal to zero. When g = 0, E = E + hrl, 7 = 7 (discussed in Chapter I). When r = 0, E = Eq I = 7 + E gl (15) 9. Load-end Quantities in Terms of Generator-end Quan- tities. — It is often desirable to express the quantities E and 7 in terms of E and 7. It is apparent that such a solution might be obtained from (14) by substituting —I for I, which would be equivalent to counting distance as positive in the direction of energy flow. To leave no doubt as to the validity of such a pro- cedure, solve the two equations (14) simultaneously for Eq and 7 . The result is _, E cosh vl — Iz sinh vl tiQ = 7 = cosh 2 vl — sinh 2 vl I cosh vl — Ey sinh vl cosh 2 vl — sinh 2 vl The denominator is seen to be equal to unity, and therefore Eq = E cosh vl — Iz sinh vl, 7 = 7 cosh vl — Ey sinh vl. (16) Other combinations derived from the original equation (14) are of interest and importance. The load at the receiving end may be represented by an equivalent resistance, or conductance, Ro — Eq 7 ' Eq = RqIq, Go = p- = 7o = GqEq, Eq (17) which values when substituted in (14) give E = Roh cosh vl + IqZ sinh vl, 7 = 7 cosh vl + IoRoy sinh vl, E = Eq cosh vl + EoGqZ sinh vl, I = GqEq cosh vl + E y sinh vl, (18) 12 DIRECT-CURRENT LINE WITH LEAKAGE from which £0 = Go cosh vl + y sinh vl ' E Ro cosh vl + z sinh vl m E cosh vl + GoZ sinh vl ' I Eq = /o = cosh vl + R y sinh vl (19) (20) 10. Voltage Regulation. — At this point the equations for voltage regulation may be developed. The regulation is defined in the same way as was done in section 3, Chapter I. By equation (14), the generator voltage, which remains constant on the removal of the load To, is E = E cosh vl + IqZ sinh vl. At no load, the conductance Go = 0, and equation (20) serves to determine the value of receiver voltage at no load with the voltage E at the generator. Let E' = no load receiver voltage; then by (20) for Go = E E cosh vl + IqZ sinh vl cosh vl cosh vl = E + Ioz tanh vl. Reg. = — „ — - = ^ z tanh vl. £j0 &Q (21) (22) For any given line, z tanh vl is a constant, and therefore for such a condition of operation that the load voltage E is maintained constant by adjustment of the generator voltage, the curve of voltage regulation when plotted to load current as abscissae will be a straight line, the same as for the simple case in which line leakage was not considered. The magnitude of the regulation will differ. The case parallel to that illustrated by equation (13), Chapter I, POWER RELATIONS IN GENERAL 13 where the generator voltage is supposed to remain constant at all loads will now be considered. E = constant generator voltage, 7 = load current. Solving (14) for the value of E in terms of E and 7 , „ E — IoZ sinh vl „ , T , . T .,__. Eq = -T-, = E sech vl — Iqz tanh vl. (23) By (20) the no-load receiver voltage is E' = E sech vl. E' — E IqZ tanh vl Reg. = Eq E sech vl — Iqz tanh vl h Ey cosech vl — I (24) Equation (24), as in the case of equation (13), Chapter I, for the line without leakage, is not that of a straight line but of a curve passing through the origin and reaching a value of infinity at a value of h = Ey cosech vl, which is the value of h occurring when the load end of the line is short-circuited. This may be verified from equation (19) by placing R = 0. 11. Power Relations in General. — At the load end of the line the power received is P = EJo watts. At any other point the power is given by P = EI, E and I being given by (14). Multiplying the two expressions in (14), P = EI = Eoh cosh 2 vl + Eolo sinh 2 vl + E 2 y sinh vl cosh vl + h 2 z sinh vl cosh vl. (25) By the hyperbolic reduction formulae in the Appendix, this reduces to P = P cosh 2 vl + \ (7 2 z + E 2 y) sinh 2 vl. (26) The total line loss is P' = P — Pa. Utilizing this relation in (25), F = P (cosh 2 vl + sinh 2 vl - 1) + E 2 y sinh vl cosh vl -f I 2 z sinh d cosh vl; P' = P 2 sinh 2 vl + {E<?y + h 2 z) sinh vl cosh vl. (27) 14 DIRECT-CURRENT LINE WITH LEAKAGE For lines with very small losses it is desirable to calculate the power loss directly by (27) since a much greater accuracy may be secured in this way than by calculating P and then numerically subtracting P , for in this latter case the difference between two large quantities of nearly the same value must be taken. In presenting the formulae as developed, it is not expected that they are to be applied indiscriminately to any and all cases, for though they are rigid, and will therefore yield accurate results wherever they are applied, it may not be at all necessary to in- troduce the degree of refinement that their use yields. Individual cases must be treated individually, if an efficient use is to be made of the material at hand. Obviously, there is little sense in calcu- lating line phenomena by rigid formulae to a degree of accuracy yielded by six-place logarithms, say, when the line constants on which the computations are based are perhaps in error by as much as two or three per cent. Such matters must be left to the judgment of the individual, but they cannot detract from the desirability and need of preparing and presenting rigid expressions. 12. Maximum Power. — Following the scheme adopted in section 3, Chapter I, the maximum value of power which can be transmitted over a leaky line by a generator of constant voltage, E, will now be investigated. To do this, E and P will be ex- pressed in terms of E and I , and since E is constant the value of I which will give a maximum P can be found. By (23) E = E sech vl — I^z tanh vl, P = E I Q = I E sech vl - I 2 z tanh vl. (28) For a maximum, the first derivative of Po with respect to I must be equal to zero, and the second derivative must be negative. dP -rp = E sech vl — 2 IqZ tanh vl = for max. (29) cPPn -Tj^ = — 2 z tanh vl, which is negative. Thus, by (29), the load current for maximum power is j 1 = Esechvl = E ^ Q) Jp„ = max. 2 z tanh vl 2 z sinh vl EFFICIENCY 15 Substituting (30) in (28) and collecting terms, E 2 E 2 ' 4 z sinh vl cosh vl 2zsinh2tfZ It is interesting to compare the expression (31) for the maximum power which can be transmitted over a leaky line with the expres- sion for the maximum power which can be transmitted over a non- leaky line, which is, by (16a), Chapter I, P 9 max.l =j E 2 4R t It is evident that 4 z sinh vl cosh vl = 2 z sinh 2 vl replaces the quantity 4 R t when there is leakage present. By introducing into (31) the condition that g = 0, the expression should reduce to (32). This will be shown to be true. Expanding 2 z sinh 2 vl into a series, using the values of z and v by (15), 2 zsinh 2 vl = 2 y -l~2 Vr^jl + ri (2 Vr~gl) z + • • • 1 = 4 rl + f r 2 gP + • • • . Now, all terms except the first contain g, and therefore for g = 0, 2 z sinh 2 v£ reduces to 4 H, which is identical with the denomina- tor of (32), since Rt = rl = total line resistance if uniformly distributed. 13. Efficiency. — In the non-leaky line, by equation (17), Chapter I, it is seen that for loads approaching zero (R approach- ing infinity) the efficiency approaches unity in value. Such is not the case for the leaky line, for it will be shown here that there is a definite limit to the possible value of line efficiency, no matter what the condition of loading may be. Using equation (26), Eff. = — = EoI ° P Eoh cosh 2 vl + \ {zU + yE 2 ) sinh 2 vl (33) 1/7 V \ cosh 2vl-\-~\z-^r-\ry-r) sinh 2 vl From (33) it is apparent that the efficiency is in no way depend- ent upon the absolute magnitude of either E or 7 but rather 16 DIRECT-CURRENT LINE WITH LEAKAGE upon the ratio of the two, that is, upon R or G . Obviously, also, the maximum efficiency will occur for such a value of this ratio as will make the term Io , Eo u = z-^r + y-y = a minimum. Letting G = tt, and remembering that y = -, we have r>o z For a minimum, from which du • 1 _ dG ~ • zG<? ~ ' 2 2 = 7T5 = Ro 2 , Z = ±/2o. <J0 In the double sign, only the positive can have a physical signifi- cance, and by investigating the value of the second derivative it can be shown that this sign gives a minimum to the value of u. Thus, for a maximum efficiency, Eo = IqZ, h = E y. (34) Substituting the ratio of E to 7o as given by (34) in equation (33), the maximum efficiency which may be attained is cosh 2 vl -\- sinh 2 vl The next and final step in this development will be to express the load current for which maximum efficiency occurs in terms of the generator voltage E, which is supposed to be constant. Introducing (23) into (34), for maximum efficiency, E sech vl — IqZ tanh vl = zlo, from which J 1 = E sech vl = -e~ vl (36) JES. . max. z (1 + tanh vl) z € Comparing (30) and (36) it is seen that the maximum efficiency does not occur coincidently with the maximum power at the load when the generator voltage is constant. NUMERICAL ILLUSTRATION 17 All of the above formulae involving hyperbolic functions could be expanded into series, thereby furnishing approximate solutions, but it is not considered at all necessary to include such develop- ments here. The above formulae, involving as they do only hyper- bolic functions of real or non-imaginary variables, are so con- veniently used numerically that to introduce series expressions leads to needless complexity. Neither does it appear necessary to introduce various approximate methods of solution, such as assuming that the entire leakage may be concentrated at the middle of the line, or the still closer approximation of placing one-sixth of the leakage at each end and two-thirds in the middle of the line. The solutions based on such assumptions are approximations, which in reality amount to using only a limited number of terms in the series expansions of the hyperbolic functions. If at any time tables of such functions are not available, it is more convenient to introduce the series for the evaluation of the hyperbolic functions than to resort to an approximate solution based on the above mentioned assumptions. 14. Numerical Illustration. — Before proceeding further with theoretical developments, a numerical illustration of the foregoing will be given. Consider a line with the following constants per mile of length. r = 50.0 ohms. g = 2 X 10~ 4 mhos. v = V7g = 0.1000. z = V- = 500.0. y = J g - = 0.00200. V g V r Let the line length be I = 15.0 miles. Then vl = 1.500. (a) What voltage would be required at the sending end to give a current of 0.050 amperes through a resistance of 200.0 ohms at the receiving end? Io = 0.050, Ro = 200.0, E = R I = 10.00. By (14), E = 10.0 cosh 1.50 + 0.050 X 500 sinh 1.50 = 10.0 X 2.35241 + 25 X 2.12928 = 76.7561 volts. (b) What is the generator current? By (14), J = 0.050 cosh 1.50 + 10.0 X 0.00200 sinh 1.50 = 0.160206 amperes. 18 DIRECT-CURRENT LINE WITH LEAKAGE (c) What is the voltage regulation under these conditions? By (22), Reg. = ^? 500 tanh 1.50 = 2.50 X 0.90515 1U.U0 = 2.263 = 226.3 per cent. (d) For a constant generator voltage of 100.0 volts, calculations of various quantities for different values of load current are given in the following table. The equations used are numbered to correspond with the text, and are given in the final numerical form for the particular case under discussion. (23) (28) (16) (21) (24) #o = 100 sech 1.5-500 J tanh 1.5 = 42.510 - 452.58 h. Pq = EqIq. I = h sech vl + Ey tanh vl, = 0.42510 7o + 0.181030. E' = 100 sech 1.50 = 42.510. E — Eo Iq Reg. = En 0.200 cosech 1.5 0.0939284 - 7 P= 100 7. Efficiency = -=£■ P' = line loss = P - P . TABLE I Calculation of Performance of Leaky Line h 0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 452.58 /„ 0.0 4.526 9.052 13.577 18.103 22.629 27.155 31.681 36.206 40.732 E a 42.510 37.984 33.458 28.933 24.407 19.881 15.355 10.829 6.304 1.778 P» 0.0 0.37984 0.66916 0.86799 0.97628 0.99405 0.92130 0.75803 0.50432 0.16002 0.4251 /, 0.0 0.00425 0.00850 0.01275 0.01700 0.02126 0.02551 0.02976 0.03401 0.03826 / 0.18103 0.18528 0.18953 0.19378 0.19803 0.20228 0.20654 0.21079 0.21504 0.21929 Reg. 0.0 0.11915 0.27053 0.46928 0.74172 1.13822 1.76843 2.92539 5.74366 22.9101 P 18.1030 18.5283 18.9532 19.3783 19.8034 20.2285 20.6536 21.0787 21.5038 21.9289 Eff. 0.0 0.02050 0.03531 0.04479 0.04930 0.04914 0.04461 0.03596 0.02345 0.00730 P' 18.1030 18.1485 18.2840 18.5103 18.8271 19.2345 19.7323 20.3207 20.9995 21.7689 The data obtained in the above table of calculations are shown plotted in Figs. 3 and 4. NUMERICAL ILLUSTRATION 19 Amp Volt 0.215 0.210 48 S\ 0.205 40 'Po 0.200 32 \ P ° 0.100 24 E> 0.190 16 0.185 8 0.180 1.0 0.8 O.fi 0.4 0.2 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Load Current I o , in Amperes Fig. 3. — Characteristics of Leaky Direct Current Transmission Line. Generator Voltage Constant, 100 Volts. Eff. 0.01 0.Q2 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Fig. 4. — Characteristics of Leaky Direct Current Transmission Line. Generator Voltage Constant, 100 Volts. 20 DIRECT-CURRENT LINE WITH LEAKAGE By equation (30) the maximum P occurs for 7 = 0.0469642 amperes, which may be noted on the plotted curve and is, by (31), 0.99821 watts. One more illustrative example will be considered. Take a line with constants, r = 5.0 ohms, and g = 2.0 X 10~ 5 mhos, per mile. Then v = 0.0100, z = 500.0, and y = 0.00200. As load values, let E = 10,000 volts and J = 10.0 amperes. For this load, the following table gives values of generator voltage, current, power, line loss, and efficiency of transmission for various line lengths. TABLE II l E / P P' Eff. miles volte amperes K.W. K.W. 10,000.0 10.0000 100.00 1.0000 20 11,207.4 14.2275 159.45 59.45 0.6272 40 12,864.4 19.0257 244.75 144.75 0.4086 60 15,038.0 24.5877 369.75 269.75 0.2705 80 17,814.9 31.1365 554.69 454.69 0.1803 100 21,306.8 38.9348 829.58 729.58 0.1205 For line lengths greater than 40 or 50 miles, the generator voltage and current, and consequently power also, are seen to increase very rapidly. The example considered is, however, an exaggerated case inasmuch as the leakage coefficient, g, is several times larger than would be encountered in an actual system which was designed to transmit power at the voltage considered. The fine constants were arbitrarily selected so as to furnish results illustrating the properties of the solutions. In both this and the former illustrative example, the fine losses were of sufficient size in comparison with the total power to be obtained readily by subtracting the power at the load from that at the generator, thus furnishing no necessity for calculating the loss directly by equation (27). The above results are shown in Fig. 5.^ 15. Effective Resistance of Line and Determination of Con- stants by Measurement. — For a direct current transmission system with no leakage the effective resistance of the line and EFFECTIVE RESISTANCE OF LINE 21 load at the generator end, as determined by the ratio E/I, is exactly equal to the load resistance plus the line resistance. For the leaky line such is not the case, the effective resistance depend- Kilo Wat Kilo Volt .... Amp , 950 19 900 18 30 800 1G 32 E/ 700 14 28 COO 12 24 1/ Kff 500 10 20 100 ■100 8 16 80 P/ 300 6 12 GO '?' 200 4 8 40 Efl 100 2 4 20 1(3 32 48 64 i Line Length in Miles 06 Fig. 5. — Characteristics of Leaky Direct Current Transmission Line. Load at Receiver Constant; E = 10,000 volts and I = 10.0 amperes. ing upon the load resistance and the two line constants r and g, as well as upon the line length. The ratio of generator voltage to generator current defines the effective resistance. Utilizing equations (18a), after canceling the common factor 7 in numerator and denominator, one has n _ E _ Rp cosh vl + z sinh vl . _. I cosh vl + R y sinh vl = effective resistance from generator end as would be indicated by a Wheatstone bridge. 22 DIRECT-CURRENT LINE WITH LEAKAGE The above equation, though complete, is scarcely as important as two special cases derivable from it. They are for the effective resistance when the load end of the line is short-circuited, and when the load end is open-circuited. The first may be derived directly from (37) by placing R equal to zero, but the placing of Ro equal to infinity (rather, allowing R to increase indefinitely) for the second case would lead to an indeterminate form. Dividing both numerator and denominator of (37) by Ro, or utilizing equa- tions (18b) gives P _ E _ cosh vl 4- Gpz sinh vl .__. / Go cosh vl -f- y sinh vl ' from which, by placing G = 0, the desired result is obtained. Thus, from (37) for R Q = 0, R' = z tanh vl = short-circuit resistance ; (39) and from (38) for G = 0, R" = z coth vl = open-circuit resistance. (40) For infinite values of line length, the two expressions (39) and (40) as well as (38) approach the value of z, since the hyperbolic tangent and cotangent both approach unity for continuously increasing values of argument. Thus, for infinite line length, the terminal or load resistance has no influence upon the effective generator-end resistance of the system, which is then equal to z = y - = infinite line resistance. (41) Equations (39) and (40) are of importance in determining the values of r and g from actual measurement of line resistance. There are two quantities to be determined, and therefore two measurements taken under different conditions will be required. The two simplest conditions to utilize are obviously those men- tioned above, viz., short-circuited and open-circuited, and from these two resistance measurements, taken either by means of a bridge or by the voltmeter-ammeter method, numerical values of R' and R" will be furnished. From these values of R' and R" the values of r and g are to be determined. EFFECTIVE RESISTANCE OF LINE Multiplying (39) and (40), r'r" = 2 2 = r, 9 and by division of these same equations, ^77 = tanh 2 vl; \ ~5Tt = tanh I Vrg. Thus, knowing the line length, I, z = J- = VWW', » n 23 v = Vrg = j tanh" 1 \-^r, r = vz = - l VrW tanh- 1 g z l VR'R" tanh \ R" (42) (43) Equations (42) and (43) suffice to accurately and completely determine the fundamental line constants from the two measure- ments, R f and R", made under the specified conditions. The above method, or its equivalent, must be used in order to deter- mine the two fundamental line constants accurately when the total line resistance and leakage are relatively large, or to be more exact, when the product of these totals exceeds a certain definite amount depending upon the accuracy desired. The ordinary approximation methods for determining the line constants consist in taking '■-i*> 9 = Rt = rl — R', 1 IR" 1 G t =gl = R'" (44) which are based on the assumptions that when the line is short- circuited for measuring R' all the current traverses the entire line — that is, is of uniform value along the line length; and that when the line is open-circuited for measuring R", the potential difference between conductors is the same throughout the length. 24 DIRECT-CURRENT LINE WITH LEAKAGE The approximation (44) is equivalent to assuming that in (43) tanh -1 y-p77 ma Y De replaced by y p77> which will obviously introduce considerable error for values of the radical greater than two-tenths. It is also apparent, by (42), that the value of the ratio K R" V R" is dependent only upon the product vl = y/R t Gt. The following table will show the magnitude of the error which the approximation method would introduce under various con- ditions: TABLE III The Inverse Hyperbolic Tangent T R" tanh"« y ~ = ^Rfit = vl |/1 ▼ R" tanh-i y j^i = ^Rfit = rf 0.00 0.00000 0.55 0.61839 0.05 0.05004 0.60 0.69314 0.10 0.10033 0.65 0.77530 0.15 0.15114 0.70 0.86729 0.20 0.20273 0.75 0.97295 0.25 0.25541 0.80 1.09861 0.30 0.30952 0.85 1.25615 0.35 0.36544 0.90 1.47221 0.40 0.42365 0.95 1.83180 0.45 0.48470 1.00 Infinity 0.50 0.54931 By the use of the above table the error introduced into the determination of the fundamental line constants by the use of the approximation method may be determined as soon as the measure- ments on which the computations are to be based are available. 16. Combinations of Leaky Lines. — In all of the preceding the discussion is applied to a line supplied with power at one end and delivering energy at the other end — a condition which per- mits the enumeration of practically all of the equations covering the phenomena of transmission of energy over leaky lines. The conditions named in the heading of this paragraph do not permit SINGLE GENERATOR SUPPLYING TWO SEPARATE LOADS 25 of such ready and general treatment, though the general method of attack may be indicated. Assuming, of course, that each individual section of line in such a system is uniform throughout its length, the fundamental differential equations (3) and there- fore the general solutions (14) still apply if the integration con- stants E and 7 are correctly interpreted. 17. Single Generator Supplying Two Separate Loads. — As a first special case, consider a generator of constant voltage E supplying power to two separate loads at A and B, the connecting transmission systems being leaky. Figure 6 serves as illustration. The two sections of line will be defined by the quantities, Section 1, r h g h l h or z h v 1} l h y x = 1/fc. Section 2, r 2 , g 2 , k, or z 2 , y 2 , h, y% = 1/%. The loads at A and B may, however, be specified in different ways. Suppose, for instance, that they be specified by means of equiva- £ Section 1. I I" Section 2. Fig. 6. — Single Generator Supplying Two Separate Loads. lent resistances, R a and Rb, respectively. To determine all currents and voltages in the system when the generator voltage is equal to E, proceed as follows: The equivalent resistance R" of the load Rb and line section 2, as measured at A, would be given by (37) as „„ _ Ea _ Rb cosh V2I2 + 32 sinh v 2 k ,.-. I" cosh vtk -+- RbV2 sinh vJq The above defined equivalent resistance is in parallel with the load resistance R a , and thus the equivalent resistance of the two combined is p, _ Ea _ RgR a ~ V - Ra + R'" G' = 1/R'. Then immediately, by (19), (46) /' = (47) R' cosh vih + 0i sinh V\li • 26 DIRECT-CURRENT LINE WITH LEAKAGE Also, Ea = R I , j E a la — q-J Ita Jfl ■"<• Jl J Again, by equation (19), J E a lib cosh ^ + z% sinh V2I2 ' Eb = Rblb- (48) (49) Finally, at the generator, by equation (12), I = T cosh vik + E a yi sinh vik. (50) Equations (45) to (50), which apply to the particular case under discussion, are arranged in convenient form for computation. They are not entirely independent inasmuch as the results obtained from one equation are introduced into one following. It is of interest to note the quantities which may be specified at will in such a transmission system. Assuming the line constants as fixed and the loading points A and B established, we may specify, for example, (1) Eb, h, and I a , (2) E b , E a , and I a , (3) R a , Rb, and E, but not such combinations as (4) E a , E b , and h, (5) E a , h, and I a , for in the latter it is readily seen that the quantities are not in- dependent. It is impossible to treat here an extended number of special cases, but the example just given may serve as a help in deciding upon a method of attack. 18. Single Load Supplied from Two Power Sources. — A different form of problem is that in which a single load is supplied from two sources of power over lines possessing leakage, as shown in Fig. 7. Again, the problem may be stated in several different ways, some of which lead to impossible solutions. For example, with SINGLE LOAD SUPPLIED FROM TWO POWER SOURCES 27 fixed generator voltage and a short-circuit at the point of loading, the current I would be the sum of the two currents supplied by the two separate short-circuited lines, which, by equation (19) when R = 0, would be E^i cosech vj,i + E 2 y 2 cosech v 2 k. It would be absurd, with fixed generator voltages, to specify a load current 7 greater than the above which could be supplied -e i' i" ^ i 9 Section 1. Section 2. Fig. 7. — Single Load Supplied by Two Generators. over a short-circuit. The same is true in the discussion of a similar system without leakage. As a concrete case, let the resistance of the load, R = -p, be jo specified. All electrical quantities for given values of E\ and E 2 are to be determined. Let the convention as to positive direction in each circuit be as indicated by the arrows. Then, with the notation shown in the diagram, i—r+r; ^ 1 (51) Eo = RoI r + RqL . J The two currents Y and I" will be taken as the two unknown quantities whose determination constitutes the main feature of the solution of this problem. By equation (14), Ei = E cosh Vik -f- I'zi sinh vh, E 2 = ^o cosh V2I2 + 1'% sinh v 2 h- Substituting for E its value as given by (51) and collecting terms, (#0 cosh Vik + Zi sinh vji) I' + (R cosh vik) I" = E h (Ro cosh v 2 k) V + (Ro cosh v^k + Z2 sinh v 2 l 2 ) I" = E 2 . (52) In (52) there are two simultaneous equations in the two unknown quantities I' and I", which suffices for their determination. The solution by determinants yields 28 DIRECT-CURRENT LINE WITH LEAKAGE ,,_ (Ro cosh ttk-l-frBmh V2U) Ex — (Ro cosh vM Ei (Ro cosh vrfi + z x sinh v x l\) (Ro cosh v t l% -+- z% sinh fj/2) — #o 2 cosh v x l x cosh y^ ' ,„_ (Ro cosh r^! +z t sinh vJi) E 2 — (Ro cosh ink) Ei (Ro cosh vJi +zi sinh i'i£i) (Ro cosh 1 j/ 2 +2jsinh »; 2 ^a) — .fto'cosh 1^ cosh ^/j ' . (z% sinh t>th) Ei -\- (z x sinh v\k) E t (Ro cosh fill + Z\ sinh v x U ) (.#0 cosh t^ + zj sinh t^ 2 ) — .fto'.cosh Vik cosh i^a (53) The voltage at the load is R I , which becomes, after expanding the denominator of the last equation of (53) and multiplying by R , En = RoIq == (22 sinh v 2 k) Ei + (z x sinh v x l x ) E 2 ._.. iZ (54) 22 cosh v\l\ sinh t^ + Zi cosh v 2 k sinh »iZi + -5- sinh fliZi sinh fl 2 k Ho 19. Unloaded Line with Double Power Supply. — Having determined the components of the load current by (53) and the load voltage, the determination of all other quantities is accom- plished with facility by the application of the several general equations applying to simple circuits. The results in equations (53) and (54) are perfectly general, holding for all values of load resistance, and they therefore lead directly to a method of deter- mining the line current and voltage at any point when no load is present by the simple device of letting the resistance R a approach infinity in value. Expanding the denominator in (53), dividing both numerator and denominator by Ro, and discarding all terms which reduce to zero on account of the infinite value of R , the resulting equations for the line currents become Ei cosh v 2 U — Ei cosh WiZi r/1 . _Uo = 00 Z2 cosh V\l\ sinh Vzk + 2 X cosh i^k sinh Vih ' Ez cosh Vik — Ei cosh vjv (55) _Uo= 00 Z2 cosh V1I1 sinh v?k + Zi cosh v?k sinh V1I1 ' while from (54), the voltage at the point in question is immediately E "I m E1Z2 sinh V2I2 + E 2 zi sinh vji . . °_Uo= °° 22 cosh Vik sinh v 2 k + 21 cosh v 2 h sinh vj,i The results expressed by equations (53) to (56) are of interest and importance. In the case of the loaded line as covered by equations (53) and (54), the distribution of current and voltage along either portion of the transmission system is conveniently NUMERICAL ILLUSTRATION 29 obtained by use of equation (14), counting distance from the point of loading and using the results obtained from (53) and (54) as initial quantities. Since the solutions as given are general, either positive or negative numerical values may be assigned to the electromotive forces E x and E 2 . The meaning of the algebraic sign in a numerical solution resulting from the use of these equa- tions follows the convention usual in any application of Kirchhoff 's Laws where a direction as indicated by an arrow is arbitrarily taken as positive. Equations (55) and (56) applied to a uniform, unloaded line serve immediately for the determination of the distribution of voltage and current. In a non-uniform or composite line con- taining two sections of line with different constants, the solution must be made first for the voltage and current at the junction point, and then, utilizing equation (14) by the method mentioned in the preceding paragraph, for any desired point throughout the length of either section. Of course, in the case of the uniform, unloaded line whose constants are the same throughout the entire length between the two sources of power supply, the lengths k and h, must be so taken that their sum is equal to the total line length. Naturally, if the line is uniform throughout, a simplified form of expression will result from either (53) and (54) or from (55) and (56), for in this case Zi = z 2 , and the denominators in (55) and (56), for example, reduce to a single function. For a uniform line whose total length is h -J- k, with constants v and z per unit length, r /l _ Ei cosh vk — Ei cosh vk Jfl 0=O o _ z sinh v (k + k) E 2 cosh vk — Ei cosh vk z sinh v (li + k) Ei sinh vk + E 2 sinh vli sinh v (l x -f- k) "'1 " (57) *1^-" The denominator is constant for all points selected, since h + k is equal to the total line length, and therefore constant. 20. Numerical Illustration. — A numerical illustration will be given here of the distribution of current and potential difference along a uniform, unloaded line, utilizing equations (57) and (58). Let the total line length be 100 miles and the line constants be r = 10.0, g = 10~ 5 , per mile. 30 DIRECT-CURRENT LINE WITH LEAKAGE Then Z = Zi = «2 = 10 3 . y = Vl = y 2 = 10- 3 . v = Vl = 2>2 = 0.0100. The denominator of (57) is 10 3 sinh 1.00 = 1175.20. The denominator of (58) is sinh 1.00 = 1.17520. Let Ei = 100 volts and E 2 = 80 volts, both positive, and therefore acting in the direction indicated by the arrows in Fig. 7. The following table contains the calculations for determining the distribution of line voltage and current. TABLE IV Line with Double Source of Power Supply h 00 10 20 30 40 50 60 70 80 90 100 h 100 90 80 70 60 50 40 30 20 10 00 *l 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 80 0.90 1.00 •1. 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Ej sinh vli 0.00 8.01 16.11 24.36 32.86 41.69 50.93 60.69 71.05 82.12 94.02 E\ sinh Wj 117.52 102.65 88.81 75.86 63.66 52.11 41.08 30.45 20.13 10.02 00.00 Sum 117.52 110.66 104.92 100.22 96.52 93.80 92.01 91.14 91.18 92.14 94.02 Eo 100.00 94.16 89.28 85.28 82.13 79.81 78.29 77.55 77.59 78.40 80.00 Ei cosh vlt 154.31 143.31 133.74 125.52 118.55 112.76 108.11 104.53 102.01 100.50 100.00 E t cosh tli 80.00 80.40 81.61 83.63 86.49 90.21 94.84 100.41 106.99 114.65 123.45 Difference 74.31 62.91 52.13 41.89 32.06 22.55 13.27 4.12 -4.98 -14.15 -23.45 10»X/' 63.23 53.53 44.36 35.64 27.28 19.19 11.29 3.51 -4.24 -12.04 -19.95 The distribution of current and voltage along the line as deter- mined in the above table of computations is shown graphically in Fig. 8. Naturally, with an unloaded line supplied with energy at both ends in the manner considered in this illustration, the direction of current flow reverses at a point coinciding with the minimum of line voltage. 21. General Network of Transmission Systems with Loads at Various Points. — Of such a system as mentioned very little can be said unless a large number of special cases be taken under consideration, and as even then the field of possibilities could in no way be covered, the solution of such problems, if an exact solution be desired, must be left to the ingenuity and mathematical ability of the individual. In general, unless the problem be stated in a manner at variance with the physical possibilities (for instance by specifying a load current at some point greater than could be supplied by generators of the selected voltage over the trans- mission systems involved), it will always be possible to obtain a GENERAL NETWORK OF TRANSMISSION SYSTEMS 31 solution. Kirchhoff's Laws will in general supply a number of current equations. If the generator electromotive forces are specified and the loads represented by their equivalent resistances, the scheme adopted in arriving at equations (51) to (54) will serve in a majority of cases. The known electromotive forces are Eo 100 Volt 98 96 94 92 90 88 86 84 I' 82 E> 80 78 76 74 72 10 Fig. 8. 20 80 90 100 I' milU- amps. 30 40 50 60 70 Distance, I z , from 100 Volt Generator. Distribution of Voltage and Current along an Unloaded Leaky Line when Supplied with Power at both Ends. E x = 100 and E 2 = 80 volts. expressed in terms of the load resistances and the unknown cur- rents. A solution of the simultaneous equations so obtained expresses the unknown currents in terms of the known generator voltages and the load resistances. When applied to a single load fed by a number of transmission systems, this procedure is very simple — leading to as many simultaneous equations as there are transmission systems involved. Figure 7 illustrates a case having two such supply systems. With the possibility of using negative values for the generator voltage's in the general solution, the generators may virtually be considered as motors (or loads) of the voltage selected, and the range of applicability of the general solution thereby extended. 32 DIRECT-CURRENT LINE WITH LEAKAGE Even with three systems of transmission involved in a network, as above mentioned, the expressions both before and after the solution of the simultaneous equations are quite complicated, and with an increase in the number of systems involved or with an addition of intermediate loads the complexity of the mathematical development increases enormously. CHAPTER III PERIODIC AND ALTERNATING QUANTITIES. FOURIER'S SERIES AND ANALYSIS OF PERIODIC CURVES 22. General. — The subject of this chapter is one which has been treated very many times. It is discussed here in order that the present volume may be rendered complete and may present to the person engaged in any investigation the means for obtaining with ease and dispatch the analysis of any given curve which is capable of being represented by a Fourier's Series. The matter of analysis by means of a finite number of ordinates, which heretofore has been considerably neglected, is also discussed. As a matter of interest a form of mechanical analyzer which was designed by the writer to facilitate the exposition of the theory of analysis is described. 23. Periodic Quantities and Fourier's Theorem. — A quantity periodic in time may be described as one which passes recurrently, in equal intervals of time, through the same series of values. An alternating quantity is a special form of periodic quantity, in that in successive equal intervals of time it passes through a series of values which are both positive and negative. A special form of Fourier's Theorem, which is sufficient for the purpose in hand, states that any continuous, single-valued, finite, and periodic function may be represented by a series of trigono- metric sines and cosines — the terms in such a series having fre- quencies which are multiples of the fundamental frequency of the periodic function. 24. Use of Fourier's Series. — The utility of Fourier's Series in the treatment of alternating current problems is based on the fact that by the separation of a periodic quantity into its constit- uent terms, a number of simple harmonically varying quantities are obtained which, in further calculation, may each be treated separately by means rigorously applicable only to such simple harmonically varying quantities. Methods of employment of the results of such an analysis will be treated in a later chapter; 33 34 FOURIER'S SERIES this chapter deals only with the methods for separating a given periodic function into the constituent terms. Usually the given periodic quantity will be specified graphically by a curve, or by a set of numerical data from which such a curve may be plotted. Thus, y may be considered as a function whose value is known for any value of 0. 25. Determination of the Unknown Coefficients by Integra- tion. — Let the scale of the independent variable be so selected that the length of the period is equal to 2 ir, and for the independent variable to this scale use the quantity or angle 0. Then we assume that y = f (0) = A + Ai cos + A 2 cos 2 + ■ ■ • + A n cos n0 + • • •+fiisin0 + £ 2 sin20 + - ■ ■ + B n sin ri0 + • • • (1)' the problem then being to determine the values of the unknown coefficients, A , A h A 2 , etc.; B h B 2 , B 3 , etc. In the usual manner, the details of which need not be discussed here, we find that by multiplying both sides of equation (1) by cos nd dd and integrating between the limits and 2 t, -If 7T Jo 2x y cos nd dd, (2) except for the special case of n = 0, when the expression 1 r 2r A = ± ydd (3) must be used. By multiplying throughout by sin nd dd and integrating, B n =-f 'yswnddB. (4) IT Jo From equation (2) it is seen that the coefficient A„ is equal to - times the area under the auxiliary curve y' = y cos n0, (n ^ 0), when measured between the limits and 2ir. Similarly, B n is er the auxiliai y" = y sin nd. equal to - times area under the auxiliary curve THE COSINE TERMS 35 26. Use of the Auxiliary Curves, y' = y cos n6 and y" = y sin n8. — If, from the original curve, these auxiliary curves are calculated and plotted, the areas enclosed could be measured with a planimeter and the values of the various coefficients thereby determined. If a number of equidistant ordinates are known for the original curve, the area of the auxiliary curves may be deter- mined approximately by taking the average height of ordinate of the auxiliary curve as computed (y' or y") and multiplying by the length of the base, 2 t. Dividing the area so obtained by r as in equations (2) and (4) and by 2 w as in equation (3), we have the simple result A = (average y), A n = 2 (avg. y cos nd), (n 9^ 0), (5) B n = 2 (avg. y sin nd), subject to the errors arising from the approximate manner of obtaining the numerical value of the integral by the use of the average value of a finite number of ordinates. 27. Determination of Coefficients from a Finite Number of Ordinates. — The determination of the various unknown coeffi- cients from a finite number of ordinates will now be considered. The given curve is to be represented by a series of the form of equation (1) in which the number of trigonometric terms may be unlimited. Divide the wave length, or length of curve which is to be rep- resented by Fourier's Series, into t equal parts, numbering the ordinates 0, 1, 2, 3, ... (t — 1), t. For the fundamental fre- quency, the total wave length is 2 tt radians, and therefore ordinate number m has an argument or angle of 2 rax t radians. y m =A Q -\-A 1 cos— — hA 2 cos2 — — (_ . . . _j_^ nC0Sn __-j_ . . . . D . 2wtt D . n 2rmr D . 2 rax , , A * -f-BiSin— — h-D 2 sin2 — - — \- • • • +B n smn— - — [- • • • . (6) I L L 28. The Cosine Terms. — If we multiply equation (6) through- out by cos p — — , where p is any positive integer, and take the summation of all ordinates over a complete wave length, 36 FOURIER'S SERIES 2rox A "g 2wt , A m 4l 2mir 2rmr cos — : — 2, j/mcosp— — = Ao v cos P —, — VM 2j cos P ~i m-l l m-l l « . . . <^\ 2wi7r 2mir . + • • • + A n 2, cosp— — cosn — f- m — 1 + #i V cos p — m-l f 2 mi . 2 rax sin ro-< 2wir . 2W7T + • • • + B n 2j cos p — — sin n — — + • • • . (7) m — 1 The summation constituting the coefficient of a general B, or sine, term may be represented by S2 rnir . 2 mir cos p — — sin n — t— > m = l ■ * which by the expansion cos a sin /3 = | [sin (a + /3) — sin (a — /S)] reduces to „„ = I "| [sin (? + f 2 "" r - sin (P^L2mj-j (g) The above summation vanishes for all integral values of p, n, and £, as can readily be seen from the fact that the summation of sines is equivalent to the summation of the projections on the vertical axis of t equal length vectors in a plane spaced under equal angles. Therefore in equation (7) there remain only the cosine, or A, terms. The summation constituting a general cosine coefficient is Zn = \% COS p 2 m-w 2 mir cosn t t which by the expansion cos a cos jS = \ [cos (a + /3) + cos (a — $)\ reduces to 1 m=l r -|X,[« . (P+")* m * + cos (p-»)2"« 1 . (9) In (9) the summation of each term taken separately will vanish unless p + n = it, or p — n — it, THE SINE TERMS 37 that is, unless n = it — p, or n = p — it, in which i represents any integral number, i must assume all values both positive and negative in order that every positive value of n be represented. Then, since 2, cos — — = t, (10) equation (7) reduces finally to m = t cy j. i = °° r- -j 5) VmCOSp -^ = 2 X A ^~P> + A (P~iO \j (11) m=l t=— oo L J in which any A with a negative subscript is meaningless and therefore to be discarded. 29. The Sine Terms. — Proceeding in a similar manner for the establishment of expressions containing only the B coefficients we multiply equation (6) throughout by sin p — — and take the t summation over a wave length. V-* . 2 rax A v^ . 2 rax . . -^ . 2rmr 2 rax £ y m sin p —r- = A Q £ sin p — \- A x £ sin p —r- cos — — m = l m = l m = l . . v^ . 2 rax 2 rax . + • • • + A n £ sin p — — cos n — — h • • • TO=1 , „ V* • 2W7T . 2W7T + £i 2, sin p — — sin — — ra = l rra=< 2 rax . 2 rax + • • • + B n 2j snip — — sinn— — + • • • . (12) 771 = 1 In a manner identical with that used for equation (8) it may be shown that the summations constituting the coefficients of the various cosine, or A, terms will all vanish separately when n and p have integral values. The coefficient of a general sine term is m=l 2 2 rax . 2 rax smp —j— sinn — — i m=l 38 FOURIER'S SERIES which by the expansion sin a sin = \ [cos (a — /3) — cos (a -f /3)] reduces to 1 V T (p — n) 2 mir (p + n) 2 mxl „ ON »» = 2 i cos j cos j ' ( 13 ) In equation (13) the separate summations entering v n are always zero unless p — n = it, or p + n = it; that is, unless n = p — it, or n — it — p, where, as before, i must take all negative and positive values in order that all positive values of n be represented. i= • • • , -3, -2, - 1, 0, +1, +2, +3, • • • Thus, by equation (10), 2 ?/ m sin p -^ = x J) B( P - it ) - B (it -p) , (14) in which, as before, negative subscripts are to be neglected. 30. Interpretation of Results — Cosine Terms. — Equation (11) may be written {=30 p ~1 O m= . t O X I ^w-p) + ^(P-fl)J = f%ymCOsp -J— (15) Obviously, the right hand member of equation (15) is twice the average value of t ordinates, equally spaced over a wave length, of the auxiliary curve y' = y cos pB, which is exactly the same as the right hand member of the second of equations (5). Consequently, it is apparent that by the use of the finite number of ordinates, t, the numerical value of twice the average ordinate of the auxiliary curve is not equal to the unknown coefficient A p alone, but to the summation of all the A coefficients of the form indicated by the left hand member of (15) which may be present in the original curve whose analysis is sought. In case an infinite number of ordinates are used, the process of averaging becomes equivalent to an integration, and thus equations (5) and (15) are equivalent for t = infinity. PARTICULAR ORDERS OF HARMONICS 39 To be specific, suppose for some particular curve we have measured 18 ordinates over a complete wave length. Also sup- pose that for the purpose of determining the cosine term of the fifth harmonic we have formed the numerical value of the right hand member of equation (15). Then, t = 18 and p = 5, so that, neglecting negative subscripts, we have A5+Ai3+A23+A31+j4 4 1+449+Ab9+ • * ' = jg 5/ Vm C ° S 5 "Ts"' Using the expression (5) for this particular case would thus give a correct value for A 6 only in the event of all the A coefficients of the orders A13, A23, A31, A41, etc., being equal to zero, that is, being absent in the particular wave shape under investigation. 31. Interpretation of Results — Sine Terms. — Equation (14) for the sine terms may be similarly treated. «=? r l rr* T 2 ^rt • 2m7r X B (p . it) - B lU - P ) \= 7 X Vm Sm V ~T' ( 16 > t'= — 00 L -1 m = l which may be compared with the third of equations (5). Applying (16) to the numerical example in the preceding para- graph, we note that B 6 - B13 + £23 - £31 + Bn - + • • • = — V y m sin 5 -==-• Similar conditions for the exact determination of B& are neces- sary here as were required for the exact determination of A 5 . 32. Particular Orders of Harmonics. — When p is zero or some multiple of -x- For the purpose of determining A we let p = 0, and then the multiplier cos p — - — in (15) is always unity. The result is then V 1 m=t or A Q -Mo + At 4- A t + An + A 2t + • • • =7 X y*» 1 «-i 1 m= t A + A t + A 2t + A zt + • • • = 7 X Vm ' ( 17 ) w=i which coincides with the first of (5) except for the disturbing terms, A t , A^, A 3t , etc. If p is an even multiple of ~ it is also a multiple of t, the case 40 FOURIER'S SERIES If p is some multiple of t, the multiplier cos p — - — again I reduces to unity and the result of the summation is the same as in (17). In regard to the sine terms, if p is zero or any multiple of t, it is obvious that the right hand member of equation (16) disappears. Building up the right hand member for p equal to t or any multiple of t, by giving negative and positive values to i, we obtain B -B +B t -B t +B 2t -B 2l + =- £ i Jm smkt^=0, (18) m — 1 where k = any integer. If p is an e just discussed. For p an odd multiple of -x, equation (15) reduces to A t + Am + Am + • • • = j^,y m cos p — - — (19) 2 2 2 l w = l l This equation reduces to A t + Au + A 5J + - • ' = -:X{-\) m y m (19a) 2 2 2 l m = l which indicates that the determination of any A coefficient whose subscript is an odd multiple of - is vitiated by the presence in the given curve of all other coefficients of the same type. Obviously, the right hand member of (16) is zero when p is an odd multiple of ~. Expanding the left hand member, we have, m when p = any odd multiple of x, B»-B t + Bu-Bu+ - • • • =j%y m smp^=0. (20) 2 2 2 2 l m = l l 33. Limitations to the Use of a Finite Number of Ordinates. — The above development as summarized in equations (15) and (16) is very important in its relation to the actual carrying out of an analysis. It is apparent that if a given curve actually consists of an infinite number of terms, or harmonics, an analysis using only SUMMARY 41 a finite number of ordinates will not give a correct result. If t ordinates are used, the lowest order harmonic which enters to vitiate the determination of the pth harmonic is that of order t — p, as long as p is less than t. Thus, if it is known beforehand that a certain curve contains harmonics up to and including a certain order, to accurately determine any or all of the cosine terms it will be necessary to use a number of ordinates at least twice the order of the highest harmonic known to be present. By equation (20) it is evident that to accurately determine all of the sine terms it will be necessary to use a number of ordinates at least one greater than twice that of the order of the highest harmonic known to be present. This arises from the fact that (20) fails to determine the B coefficient of order =• 34. Summary. — Although, as was pointed out in the pre- ceding paragraph, it is impossible to secure a rigorously true analysis of a curve containing an infinite number of harmonics by the use of a finite number of ordinates, this fact need cause the engineer, interested primarily in results accurate only within the limits set by the possibilities of physical measurements, much concern. As a rule, in any curve representing physical phenomena the magnitude of the various harmonics present in a periodic curve decreases with increase in order, and if so, it will usually be possible to use a sufficient number of ordinates to cause equations (5) to be sensibly true. Though there is no intent of discussing here the question of convergence of Fourier's Series or the many very interesting properties of such, it may be well to remark that by the selection of a certain number of ordinates covering one wave length, a similar number of points are established on the original curve through which the curve represented by Fourier's Series resulting from the analysis will pass. This will be true when the subscripts to the A or B terms are taken to a number not to exceed ~ , when the ordinates are numbered according to the scheme outlined. Thus by selecting, say, 36 ordinates (t = 36), by equations (15) and (16) we may determine A , A h A 2 , . . . An, and Ai 8 , as well as Bi, Bi, . . . Z?ie, and Bn, 42 FOURIER'S SERIES making in all 36 independent coefficients determined from the 36 ordinates. As pointed out in section 32, equation (20), it is impossible to determine B^ in this case. The Fourier's Series thus established will pass exactly through the 36 points selected, so that the only possibility for difference between the original curve and that resulting from the analysis is in the shape of the arch between ordinates. Again, if 35 ordinates equally spaced over a wave length were used, values of A , Ai, A if . . . A i6, and An, B h B i} . . . Bi 6 , and B„ would be determined — this time 35 independent coefficients from the 35 ordinates — which would pass a curve exactly through the 35 points. The two sets of coefficients would in general be differ- ent, depending upon the magnitudes of the harmonics higher in order than 17 present in the original curve. As far as engineering and physical sciences are concerned, analysis with a finite number of ordinates can be made sufficient, for in a majority of cases the original curve is specified only by a number of plotted points — the shape of the arch between points being unknown at best. In such cases an analysis as above indi- cated will give the simplest curve which will represent all of the data. Again, in the analysis of curves which are drawn continu- ously by some recording device, the accuracy of an analysis will in general be affected more by the inability to secure perfect measure- ment of instantaneous values than by the failure of the methods given to yield exact results; Illustrative examples of analyses will be shown later. 35. Mechanical Analysis Based upon Integration Processes. — The planimeter is a device which gives directly the numerical value of an integral as typified by the area enclosed by a curve. What a mechanical analyzer must accomplish is to give, by means of a planimeter, the numerical values of integrals of the form indi- cated in equations (2), (3), and (4). This has been done in several ways, one of which is described here. The apparatus is illustrated schematically in Fig. 9. In the drawing, y = f (x) is the given curve to be analyzed, and to fix our ideas let us suppose that the coefficient B n is desired. T is a tracing point with which to follow the given curve. A is the main body carrying the tracing point, which is constrained to a MATHEMATICAL THEORY OF ANALYZER 43 parallel motion by the mechanism B. A rack, at the end of which is a slide cross-head, is free to move backwards and forwards through the fixed guide G. Thus, the movement of the rack is exactly equal to the x movement of the tracing point T. Place planimeter point in Intersection of_two right angle slots, P. Fig. 9. — An Harmonic Analyzer. The motion imparted by the rack to the gear D is transmitted through the gear-train M to the face-plate in which is a projecting pin at a distance R from the center as shown. Let the gear-train be so dimensioned that the radius R makes n revolutions to one movement of T from to 2 t (one wave length) on the x axis of the given curve. The action of this pin in the slotted cross-head gives a horizontal movement to the bar E dependent upon either the sine or cosine of n times the horizontal displacement of the tracer T. To the body A is attached the slotted arm H which moves in a vertical direction an amount equal to y on the given curve. Therefore a planimeter point placed in the right-angle intersection of the slots in H and E will have a motion made up of the two components as described. 36. Mathematical Theory of Analyzer. — Take supplementary axes as shown, corresponding to the center of the circle described 44 FOURIER'S SERIES by R and to the position occupied by the slot in A when the point T is on the x axis of the given curve. Let the radius R be in its extreme position to the left when T is at the origin of the given curve. Denote coordinates of the point P with reference to the supple- mentary axes by x' and y'. Then y' = y, x' = — R + {R — R cos 0) = —R cos <£, where </> is the angular displacement of the radius R in a clock- wise direction from its initial position to the left. But <j> = nx, and thus x' = —Rcosnx, y' = If. (21) In the new curve whose area may be denoted by S', dS' = y' dx' = y (Rn sin nx) dx, y sin nx.dx. (22) But from equation (4) 1 C 2r B n = - I ysinnx dx, IT J0 so that by comparison with equation (22) where S' is the actual area which would be indicated by the planim- eter whose tracing point follows the intersection of the two slots in the moving arms. In a similar manner it can be shown that A " = &n' CM) where S" is the area indicated by the planimeter when the initial position of the radius R is taken in the vertical position above the center. Obviously, according to equation (4), A would be determined by passing the planimeter over the original curve and dividing the area so obtained by the length of base. SELECTION OF THE NUMBER OF ORDINATES 45 Assuming the apparatus to function perfectly, such a device would yield exact results regardless of the number of harmonics present in the wave. Though of interest, and of value in obtaining a clear idea of the theory of analysis, such a mechanical scheme is not very satisfactory for actual use, in that the labor required to obtain a complete analysis with such a device would be much greater than for procedures based on the measurements of a finite number of ordinates; particularly if tables are available to facili- tate the latter form of procedure. Another disadvantage of most mechanical analyzers is that the original curve must be drawn to such a scale that one period or cycle of the fundamental frequency has a certain definite wave length. 37. Tables and Forms for Analysis from a Number of Ordi- nates. — In regard to analysis by means of a number of ordinates, the various schemes in use are in reality only methods for obtaining rapidly the numerical values of the summations indicated by the right hand members of equations (15) and (16). Of course it is possible to determine any particular coefficient, for instance A p , by taking every one of the t ordinates and after multiplying it by the cosine of p times its angle to add it with all similar products from the other ordinates to obtain the summation. This process would have to be repeated for every coefficient desired, and it is apparent that a great amount of time and effort would be consumed, particularly if many ordinates are used. After completing an analysis in such a manner it would be noticed that a great many multiplications involving the same multiplier and multiplicand had been performed. It is the aim of the various tabular schemes in use to minimize the labor of an analysis by eliminating as many as possible of such duplicate operations. 38. Selection of the Number of Ordinates. Separation of Harmonics. — It will in general be advisable to use an even number of ordinates, for then the ordinates of the second half-wave will be located at points on the axis differing by 180 degrees from the abscissae used in the first half-wave. This permits of the separation of the odd and even harmonics, as will be pointed out. Since in a majority of cases connected with electrical engineering problems the periodic waves encountered will not contain the harmonics of even order, it is desirable to arrange an analysis scheme in such a way as to make possible the determination of the 46 FOURIER'S SERIES odd harmonics only, but at the same time to provide means for a complete analysis, should such be desired. Since [— cos p/3 when p is odd and also COS p(P + ir) = , cos p/3 when p is even fa . % (— sm P/3 when p is odd Sinp(j8 + 7r) = \ • u - sin p/3 when p is even (25) it is evident that by taking the average value (one-half the alge- braic sum) of two ordinates separated by one-half wave length, or 7r, we will have the value of that portion of an ordinate which is due to even harmonics only. Similarly, by taking one-half the difference of two ordinates separated by one-half a wave length we obtain the ordinate due to odd harmonics only. Thus Vm" = \ \Vm + yit \~l = ordinate due to even harmonics only. (26) L \2 + m )\ Vm = h Yl)m — V/t \~| = ordinate due to odd harmonics only. (27) L l2 +m ;J This separation of the given data into two sets of data, one containing only even harmonics and the other only odd harmonics, is made possible by the use of an even number of ordinates, t. 39. Description of Tabular Forms for Complete Analysis. — These forms are based on summations of the types (15) and (16), the summation being carried from m = to m = t — 1, instead of from m = 1 to m = t. The result is of course the same, but a simplification in notation is obtained which makes the change desirable. The description of the forms follows the order of use in making an analysis from 72 ordinates measured over a complete wave, but as will be apparent, it is not always necessary to use such a large number of ordinates. In fact the forms are arranged so as to accommodate an analysis for both odd and even harmonics from either 72, 36, or 18 ordinates per complete wave, or an analysis for odd harmonics only from either 36 or 18 ordinates measured in a half-wave. The given data, 72 ordinates per wave, are first separated into two sets of data by equations (26) and (27), one set containing only odd and the other only even harmonics. Form A at the end of the volume is used for this separation. DESCRIPTION OF TABULAR FORMS 47 The 36 values of y' obtained from Form A are then to be used in ^orm B as original data, and from the sums and differences formed there the odd harmonics are determined and checked by Forms C and D. The process of multiplication by sine or cosine of pd is carried out in Form C, while in Form D a number of ordinates are actually computed from the resulting equation, which by coincidence with the original, measured ordinates, estab- lishes the correctness of the numerical computations. Form B is provided with a table for the collecting of the entire results of the analysis. The 36 values of y" covering a half wave length of the funda- mental frequency still remain, and from these the even harmonics are to be determined. This set of data is equivalent to a set of 36 ordinates covering a complete wave length of the second har- monic, which may now be regarded as the fundamental frequency in a new set of harmonics whose frequencies are all multiples of two times the frequency of the original fundamental. Thus the even harmonics, 0, 2, 4, 6, 8, ... , etc., in the original wave become harmonics of order 0, 1, 2, 3, 4, ... , etc., in the new system whose fundamental wave length is one-half that of the original. In the new system, the even harmonics may be separated from the odd in exactly the same manner as was done in Form A for the original system, by combining ordinates separated by one-half of the new wave length. The entire new system is equivalent to a set of 36 ordinates over a complete wave, and the tables are therefore prepared with a notation corresponding to such a measurement of original data, and are described accordingly. It must be remembered that if it is an analysis of a new derived system from 72 ordinates per complete wave which is carried out, the actual frequencies will be twice those indicated by the tabular results — in other words, A 2 , Ae, Ai , Au, etc., instead of A h A 3 , Ai, A 7 , etc., as indicated in the tabular notation of Forms F and G. Form E is to be used for separating the set of 36 ordinates per wave length into two sets of 18 ordinates per half wave length, one set containing only the odd and the other only the even har- monics. The 18 values of y' from Form E are to be used in Forms F and G for the determination and checking of the odd harmonics. If it is known that the original wave contains no even harmonics, Forms F and G will serve immediately for the determination of 48 FOURIER'S SERIES the odd harmonics from 18 ordinates measured over a half -wave. Forms B, C, and D may also be used for this purpose with the exception that 36 ordinates per half-wave are used, thus giving a greater accuracy than that obtained with 18 ordinates. After having utilized the 18 values of y' from Form E, the 18 values of y" still remain, which cover a quarter wave length of the 72 ordi- nate curve, or a half wave length of the derived 36 ordinate curve, and which are made up from the harmonics of order 4, 8, 12, 16, etc., of the original 72 ordinate curve, or from the harmonics of order 2, 4, 6, 8, etc., of the new 36 ordinate curve. These eighteen ordinates thus represent a full wave length of a periodic curve whose fundamental frequency is four times that of the original 72 ordinate curve, or twice that of the 36 ordinate curve. Forms H and I are arranged for the analysis of such a full wave length represented by 18 ordinates. Obviously, if the analysis is based upon y" from Form E, the frequency of the resulting terms must be multiplied by four to be put into the series represented by the original data of Form A or partially expressed by the analysis for odd harmonics in Forms B, C, and D. Form H as given here is substantially a reproduction of a form or schedule given by Grover in " Bulletin of the Bureau of Stand- ards," Vol. 9, page 646. The reader is referred to this bulletin for a partial bibliography of the subject, and for schedules or forms for analysis to be used with a number of ordinates still smaller than eighteen per complete wave, as covered by Form H. 40. Summary. — To summarize : the forms described provide for a complete analysis for both odd and even harmonics including a constant term, from 72 ordinates per complete wave. To ac- complish this end, all of the forms are used — the odd harmonics being determined first, then those whose frequencies are twice those of the odd harmonics, and finally the constant term and those harmonics whose frequencies are four times the odd harmonic frequencies. In the use of Form I, it is of course to be noticed that twice the values of A and A 9 are obtained, as signified in the notation at the bottom of the blank form for analysis. This comes about as a result of equations (17) and (19), in which the factor 1/t is used in place of the factor 2/t. If desired, the odd harmonics up to and including the 35th may be determined from a measurement of 36 ordinates over a half NUMERICAL ILLUSTRATION 49 wave as original data, if it is sufficient to represent this half wave only or if it is definitely known that no even harmonics exist in the wave. Also, a complete analysis from 36 ordinates per full wave or an analysis for odd harmonics only from 18 ordinates per half wave is provided for, and finally an analysis for odd and even harmonics from 18 ordinates per full wave. If in any case the determination of but one particular harmonic is desired, it is of course possible to omit all of the computations X 2TT Fig. 10. — The Broken-line Periodic Curve Represented by Equation (28). except those directly involved in the various steps leading thereto. A little experience in the use of the forms will immediately suggest simplifications of procedure which may be possible in special cases. Analysis forms of the type shown here have been in constant use by the writer, and in all respects have been found the most satisfactory means for separating a periodic curve into its con- stituent harmonics. 41. Numerical Illustration as Check on Tabular Forms. — In order to check the analysis forms as well as to furnish an illustration 50 FOURIER'S SERIES of what might be expected in the way of accuracy from their use, an analysis of a curve whose constituent harmonics could be determined by integration was made. The curve used is made up of a number of straight lines whose equations are known so that the numerical values of the ordinates may be computed with accuracy. The curve is made up of the sum of two curves, one containing odd and the other even harmonics, so that both odd and even harmonics are present in the resultant. Fig. 10 shows the wave shape resulting from the sum of the two component curves. Its equation, when x is measured in radians, is y -y +y = ^ ~ *[cos(x + l)+^cos3(x+l) +^cos5(z+l)+ • • -] + |~ -rcos2(x-0.2) + icos6(x-0.2) + ^coslO(a;-0.2)H 1 (28) ir [_ y Zo J Expanding cos (x + a) by cos a cos x — sin a sin x, the A and B coefficients are obtained, and the complete curve, y = / (x), is represented by a Fourier's Series in which the coefficients have the values given in Table V. From the known line slopes and intercepts the 72 ordinate values as given in Table VI were obtained. These ordinates were equally spaced over a wave length. NUMERICAL ILLUSTRATION 51 TABLE V Coefficients of Fourier's Series Representing the Broken Line Curve. (Equation 28.) Odd harmonics Even harmonics n A n B n n A n B n 2 6 10 +3.141593 -1.17273 -0.051264 +0.021194 1 3 5 -0.68793 +0.14005 -0.014447 +1.07140 +0.019966 -0.048837 -0.49581 -0.131859 -0.046310 7 9 11 13 -0.019590 +0.014322 -0.00004657 -0.0068366 +0.017071 +0.0064781 -0.010522 +0.0031655 14 18 22 26 +0.025622 +0.014096 +0.0032343 -0.0035298 -0.0043209 +0.0069560 +0.0100132 +0.0066560 15 17 19 21 +0.0042990 +0.0012120 -0.0034871 +0.0015814 +0.0036798 -0.0042356 +0.00052861 +0.0024155 30 34 • 38 42 -0.0054335 -0.0038303 -0.00088618 +0.0014993 +0.0015811 -0.0021768 -0.0034138 -0.0024673 23 25 27 29 31 +0.0012824 -0.0020192 +0.00051023 +0.0011325 -0.0012120 -0.0020367 -0.00026962 +0.0016703 -0.0010047 -0.00053532 46 50 54 58 62 +0.0023463 +0.0017093 +0.00033942 -0.00086036 -0.0013066 -0.00053648 +0.0011082 +0.0017132 +0.0012457 +0.00021941 33 35 37 39 +0.00001552 +0.00093928 -0.00071187 -0.00022322 +0.0011691 -0.00044503 -0.00059852 +0.00080679 66 70 74 78 -0.00094224 -0.00014212 +0.00057232 +0.00083222 -0.00069223 -0.0010296 -0.00073312 -0.00009020 41 43 45 47 +0.00074782 -0.00038225 -0.00033030 +0.00057196 -0.00012015 -0.00057277 +0.00053501 +0.00007122 82 86 90 94 +0.00058318 +0.00005418 -0.00041517 -0.00057568 +0.00048332 +0.00068647 +0.00047218 -0.00002855 49 51 53 55 -0.00015940 -0.00036330 +0.00041626 -0.00000931 etc. -0.00050577 +0.00032809 +0.00017947 -0.00042079 etc. 98 102 106 110 -0.00038785 -0.00000996 +0.00031867 +0.00042088 etc. -0.00036164 -0.00048942 -0.00032234 +0.00000373 etc. 52 FOURIER'S SERIES TABLE VI Seventy-two Ordinates for Check Analysis of Periodic Curve m r» m Y m m Y m m Y m 1.40000 18 5.31239 36 2.54159 * 54 3.31239 1 1.31274 19 5.57419 37 2.27980 55 3.39966 2 1.22546 20 5.83599 38 2.01799 56 3.48692 3 1.38540 21 5.85059 39 2.00339 57 3.32699 4 1.64720 22 5.76332 40 2.09066 58 3.06519 5 1.90899 23 5.67606 41 2.17792 59 2.80339 6 2.17080 24 5.58880 42 2.26519 60 2.54160 7 2.43260 25 5.42138 43 2.35246 61 2.35993 8 2.69439 26 5.15959 44 2.43972 62 2.27267 9 2.95620 27 4.89779 45 2.52699 63 2.18540 10 3.21799 28 4.63598 46 2.61426 64 2.09813 11 3.47979 29 4.37419 47 2.70152 65 2.01086 12 3.74160 30 4.11239 48 2.78880 66 1.92360 13 4.00339 31 3.85058 49 2.87606 67 1.83633 14 4.26519 32 3.58879 50 2.96332 68 1.74906 15 4.52699 33 3.32699 51 3.05059 69 1.66180 16 4.78879 34 3.06520 52 3.13786 70 1.57454 17 5.05059 35 2.80339 53 3.22512 71 1.48726 The above table contains the ordinates for each five degrees along the axis of x. These ordinates represent the values to which the infinite series (28) would converge for the assumed abscissae. From the data given in Table VI a complete analysis was made, using all of the forms. The values of coefficients derived thereby are given in Table VII. The results of the complete analysis as tabulated in Table VII are to be compared with the corresponding values in Table V, which contains the true values of the coefficients. The differences noted arise from the effect of the disturbing terms in the sum- mation-equations (15) and (16). For example: the true value of A 2 is — 1.17273, while from the analysis we obtain A 2 = — 1.17205, in this case rather close agreement, but still not exact. By equa- tion (15), however, the value —1.17205 should be equal to the sum of the true values of A 2 + An + A 7 * + Am + Ai M + • • • • Forming this sum partially from Table V, At + A 70 + A 7i = - 1.17230, COMPARATIVE ACCURACY 53 a value nearer the result of analysis than the actual value of A 2 , but still differing therefrom on account of the neglected terms in the infinite series, A 14 2 + ^146 + A 2 14 + A218 + ' ' * . Again, by equation (16), using the true values from Table V, B 2 - £ 70 + B u = -0.49551, a quantity very nearly equal to —0.49543, which is the value of Z? 2 resulting from the analysis. TABLE VII Coefficients of Fourier's Series Derived by Complete Analysis from 72 Ordinates per Complete Wave, Fig. 10 H ■"» B n n A n B n 2 3.14159 -1.17205 1 -0.68772 1.07106 -0.49543 3 0.13966 0.01992 4 5 -0.01432 -0.04846 6 -0.05125 -0.13102 7 -0.01930 0.01679 8 9 0.01393 0.00633 10 0.02040 -0.04578 11 -0.00001 -0.01009 12 13 -0.00645 0.00294 14 0.02344 -0.00912 15 0.00392 0.00342 16 17 0.00113 -0.00376 18 0.01373 0.00566 19 -0.00300 0.00039 20 21 0.00123 0.00201 22 0.00419 0.00869 23 0.00105 -0.00152 24 25 -0.00141 -0.00028 26 -0.00153 0.00651 27 0.00022 0.00109 28 29 0.00069 -0.00045 30 -0.00368 0.00334 31 . -0.00046 -0.00036 32 33 -0.00016 0.00035 34 -0.00403 0.00094 35 0.00020 0.00011 36 An inspection of the values of the small coefficients of the high frequency terms will reveal a very considerable error in the analy- sis. These errors arise from the presence in the original curve of an infinite number of harmonics. In the analysis of any experi- mentally obtained curve such a condition would not be expected to occur, and therefore the results would not be so much in error. In all events, the 72 points selected on the original curve are represented by the resulting equation. 42. Comparative Accuracy. — To illustrate the deviation of the curve obtained by analysis, from the original curve, an analysis 54 FOURIER'S SERIES of the preceding broken line curve was made from 18 ordinates (that is, every fourth ordinate as tabulated in Table VI was used in Forms H and I as original data) with the results shown in Table VIII. TABLE VIII Coefficients of Fourier's Series Derived by Complete Analysis from 18 Ordinates per Complete Wave, Fiq. 10 n A n B n n A H B n 3.15533 -0.68939 -1.17608 0.14465 0.02762 5 6 7 8 9 -0.02018 -0.05493 -0.02003 0.01886 0.01415 -0.05255 1 2 3 4 1.07510 -0.49637 0.01817 0.01780 -0.13436 0.02705 0.05229 Again, for illustration, by equation (16), using true values of coefficients from Table V, and for t = 18 and p = 6, Be — Bn + -B24 — .B30 4" B& ~ B& + Bw — Bm -f- Z?78 — B& "T* -^96 ~~ B102 = —0.13482, which is nearly equal to the value —0.13436 as given by the 18 ordinate analysis for the coefficient B 6 . Figure 11 shows the original broken line curve and also the curve represented by the finite Fourier's Series having the coeffi- cients given in Table VIIL The eighteen ordinates from which the analysis was made are shown in the figure, and corresponding ordinates in both curves should theoretically be equal. The lower, smooth curve, represented by the finite series containing terms up to the ninth harmonic was drawn by means of a curve tracing and analyzing machine designed and constructed by Pro- fessor J. N. LeConte, of the University of California. Small differences in the corresponding ordinates of the two curves may be attributed to slight errors in adjustment of the curve tracing apparatus. The approximation of the derived curve to the original is clearly shown, nevertheless. In order to facilitate the computation of ordinate values and the plotting of wave shapes from Fourier's Series with given coefficients by those who are not fortunate enough to have a curve-tracing device at hand, Table VIII A has been prepared, from which COMPARATIVE ACCURACY 55 the trigonometric functions of multiple angles (up to and including the 36th harmonic) may be obtained by finding the function of some angle included in the first quadrant. The upper algebraic sign is for cosine and the lower algebraic sign for sine functions. 27T N^ f1\ 2ir Fig. 11. — The Original Curve and the Curve Represented by the Results of an Eighteen Ordinate Analysis. TABLE VIII A Table for Trigonometric Functions of Multiple Angles ss— tea* Values of <t> tat ■B Example 23 X 75° - cos 75° 23 X 75° - -sin 75° 1 2 3 4 5 6 7 8 9 5 5 10 15 20 25 30 35 40 45 10 10 20 30 40 50 60 70 80 90 15 15 30 45 60 75 90 =F75 =F60 T45 20 20 40 60 80 =F80 =F60 =F40 T20 - 25 25 50 75 =F80 T55 =T=30 T 5 120 I« 30 30 60 90 =F60 T30 - 130 160 -90 35 35 ,70 =F75 =F40 T 5 130 "65 ±80 ±45 40 40 80 =F60 T20 ~20 160 ±80 ±40 + 45 45 90 =F45 - Z« -90 ±45 + ft 50 50 =F80 T30 ~20 170 ±60 ±10 +40 +90 55 55 =F=70 =F15 ~40 ±85 ±30 ft +.80 T45 60 60 =F60 - ~69 ±60 + +.60 T60 - 65 65 =F50 115 180 ±35 +.30 T85 =F20 145 70 70 =F40 130 ±80 ±10 +.60 =T=50 ~20 -90 75 75 =F30 145 ±60 ft +90 T15 ~60 ±45 80 80 =F20 ~60 ±40 ft =F60 120 ±80 + 85 85 =F10 175 ±20 ^65 T30 155 ±40 +45 90 90 - -90 + +90 - -90 + +90 95 T85 Z io ±75 +.20 =F65 130 ±55 ft =F45 100 =F=80 ~20 ±60 +40 =F40 160 ±20 +.80 - 105 =F75 130 ±45 +.60 =F15 -90 +15 T60 I« 110 =F70 ~40 ±30 ^80 Iio ±60 +.50 T20 -90 115 =F65 150 ±15 T80 135 ±30 > 120 ±45 120 =F60 ~60 + T60 160 + =F60 ~60 + 125 T55 ~70 iu =F40 185 ^30 T25 ±80 ft 130 T50 ^80 +.30 =F20 ±70 +.60 Iio ±40 +90 135 =F45 -90 ft - ±45 +90 145 + =F45 140 =F40 ±80 +.60 120 ±20 =F60 180 ft - 145 T35 ±70 ft 1*0 ft =F30 ±65 +.80 1*5 150 =F30 ±60 +90 160 ^30 - ±30 T60 -90 155 =F25 ±50 =F75 180 i 55 130 + 5. =F20 ±45 160 =F20 ±40 T60 ±80 +.80 ~60 i* ~20 + c 165 =F15 ±30 =F45 ±60 T75 -90 +75 160 +45 170 =F10 ±20 =f30 ±40 T50 ±60 =F70 ±80 +90 175 T 5 ±10 T15 ±20 =F25 ±30 T35 ±40 =F45 180 - + - + - + - + - Page 56 TABLE VIII A — (Continued) Table for Trigonometric Functions op Multiple Angles 9^ 10 11 12 13 14 15 16 17 18 5 50 55 60 65 70 75 80 85 90 10 =F80 =F70 =F60 =P50 =F40 =F30 =F20 T10 - 15 T30 T15 - 115 130 145 160 175 -90 20 ~20 ~40 160 180 ±80 ±60 ±40 ±20 + 25 ~70 ±85 ±60 ±35 ±10 t» +.40 +.65 +90 30 ±60 ±30 + +.30 +.60 +90 T60 T30 - 35 ±10 +.25 +60 =F85 T50 T15 120 155 -90 40 +.40 +.80 T60 =F20 120 160 ±80 ±40 + 45 +90 =F45 - ~45 -90 ±45 + +45 +90 50 T40 Iio ~60 ±70 ±20 +.30 +.80 =F50 - 55 1 10 ~«B ±60 ± 5 +.50 T75 =F20 135 -90 60 ~60 ±60 + +60 =F60 - 160 ±60 + 65 ±70 ±5 +.60 T55 1 10 175 ±40 +25 +90 70 ±20 +50 =F60 Iio 180 ±30 +.40 =F70 - 75 +.30 T75 - 175 ±30 +45 =F60 115 -90 80 +.80 =F20 ~60 ±40 +«> =F60 120 ±80 + 85 =F50 135 ±60 +.25 =F70 115 ±80 + 5 +90 90 - -90 + +90 - -90 + +90 - 95 ~50 ±35 +.60 T25 170 ±15 +.80 =F 5 -90 100 ±80 +a> =F60 ~40 ±40 +.60 =F20 180 + 105 ±30 +75 - ±75 +.30 T45 160 ±15 +90 110 £20 =F50 ~60 ±10 =F80 130 ±40 +.70 - 115 +.70 ~ 5 ±60 +.55 =F10 ±75 +40 T25 -90 120 T60 ~60 + =F60 Zoo + =F60 Zoo + 125 =F10 ±65 +60 I 5 ±50 > 120 ±35 +90 130 ~40 ±10 =P60 ~70 +20 =F30 ±80 +.50 - 135 -90 +.45 - ±45 +90 ~45 + =F45 -90 140 ±40 =F80 160 +20 T20 ±60 +.80 140 + 145 > =F25. ±60 +85 1 50 +15 =F20 ±55 +90 150 +.60 ~30 + =F30 ±60 +90 160 +30 - 155 =F70 ~85 +.60 135 t„ T15 ±40 T65 -90 160 =F20 ±40 =F60 ±80 +.80 160 + 40 ~20 + 165 130 +15 - ±15 T30 ±45 =F60 ±75 +90 170 180 +.70 160 +^50 "40 +.30 120 t» - 175 ±50 =F55 ±60 =F65 ±70 =F75 ±80 T85 -90 180 + - + - + - + - + Page 57 TABLE VIII A — (Continued) Table for Trigonometric Functions of Multiple Angles ^v? • ^ 19 20 21 22 23 24 25 26 27 5 T85 =F80 T75 =F70 +65 T60 =F55 =F50 T45 10 Zio 120 130 1*0 150 ~60 ~70 Z80 -90 15 ±75 ±60 ±45 ±30 ±15 + f +J30 %m 20 > p. +.60 +.80 T80 TOO =F40 =F20 - 25 =F65 =F40 =F15 Z io 135 Zoo Z85 ±70 ±45 30 130 160 -90 ±60 ±30 + o +.30 +.60 +90 35 ±55 ±20 > ^50 +"85 =F60 =F25 Zio Z« 40 45 =f=45 +.80 - =F60 ~45 =F20 -90 ~20 ±45 ~60 + ±80 > ±40 +90 + =F45 50 150 ±80 ±30 ^20 + 70 =F60 =F10 140 -90 55 ±35 +.20 p, T50 Z 5 loo ±65 ±10 *■ 60 +.eo T60 - ~60 ±60 + +.60 =F60 - 65 =F25 1*0 ±75 ±10 +.55 T60 Z 5 Z70 ±45 70 ~70 ±40 +.30 =F80 T10 Zoo ±50 +.20 +90 75 ±15 +.60 =F45 130 ±75 + +75 =F30 ^45 80 +.80 =F20 160 ±40 + 40 =F60 ~20 ±80 + o 85 =F 5 180 ±15 +.70 =F25 Zoo ±35 +.50 T45 go -90 + +90 - -90 + +90 - -90 95 t« =F80 Z 1 * ±70 ^25 =F60 Z35 ±50 > 100 =F80 120 ±60 t.40 T40 Zoo ±20 +.80 - 105 Zu ±60 t" T30 ~75 + =F76 Z30 ±45 110 ±70 *■ =F30 "80 p. =F60 Z50 ±20 +90 115 +25 =F40 175 > =F55 Zoo ± 5 =F70 145 120 =T=60 ZOO + =F60 160 + o T60 ~60 + 125 135 ±20 =F75 150 ± 5 =F60 ~65 +,. =F45 130 ±50 ^80 130 ±20 =F70 Zoo > =F40 -90 135 +45 - ±45 +90 Z<8 + =F45 -90 +45 140 =F40 ±80 +.60 ~20 ±20 =F60 ^80 + 40 - 145 155 +.20 =F15 ±50 T85 Zoo +J25 T10 ±45 150 ±30 =F60 -90 +.60 130 + =F30 ±60 +90 155 +.65 "40 > =F10 ±35 T60 ±85 +.70 "45 160 =F20 ±40 =F60 ±80 +.80 Zoo + 40 "20 + 165 175 +.60 ~45 ^30 in + T15 ±30 =F45 170 ±10 =F20 ±30 =F40 ±50 T60 ±70 =F80 -90 175 > 180 > 170 +.65 Zoo +65 ~50 > 180 - + o - + - + - + o - Page 58 TABLE VIII A— (Concluded) Table for Trigonometric Functions of Multiple Angles 8 28 29 30 31 32 33 34 35 36 5 =F40 =F35 =F30 =F25 T20 =F15 =F10 T 5 -0 10 ±80 ±70 ±60 ±50 ±40 ±30 ±20 ±10 +0 15 +«) p. +90 =F75 T60 =F45 =F30 T15 -0 20 ~20 _40 160 180 ±80 ±60 ±40 ±20 +0 25 ±20 + 5 + 6 +30 +.55 +.80 =F75 =F50 =F25 -0 30 =F60 =F30 - "30 _ 60 -90 ±60 ±30 +0 35 180 ±65 ±30 I' > P> T70 =F35 -0 40 +.40 +.80 =F60 =F20 120 ~60 ±80 ±40 +0 45 - ~45 -90 ±45 + +45 +90 =F45 -0 50 ±40 t» +.60 =F70 =TF20 ~30 180 ±50 +0 55 =F80 =F25 130 185 ±40 +15 +.70 T55 -0 60 160 ±60 + +« =F60 - 160 ±60 +0 65 +.20 +.85 =F30 ~35 ±80 ±15 +.50 =F65 -0 70 _T20 ~50 ±60 +10 +.80 =F30 ~40 ±70 +0 75 ±60 +15 +90 =F15 ^60 ±45 +.30 =F75 -0 80 +.80 =F20 160 ±40 +.40 =F60 ~20 ±80 +0 85 1*0 ±55 +.30 =F65 120 ±75 +10 =F85 ' -0 90 + +90 - -90 + +90 - -90 +0 95 =F40 155 ±30 +.65 =F20 175 ±10 +.85 -0 100 ±80 +.20 =F60 ~40 ±40 +.60 =F20 ~80 +0 105 +.60 =F15 -90 +15 =F60 ~45 ±30 +75 -0 110 ~20 ±50 +.60 =F10 ±80 +M> =F40 170 +0 115 ±20 T85 130 ±35 +.80 115 ±50 +.65 -0 120 =F60 ~60 + =F60 160 + =F60 ~60 +0 125 ~80 +25 =F30 ±85 +.40 T15 ±70 +55 -0 130 +.40 =T=10 ±60 +70 ~20 ±30 =F80 "50 +0 135 - ±45 +90 ~45 + T45 -90 +45 -0 140 ±40 T80 160 +.20 =F20 ±60 ^80 140 +0 145 =F80 165 +.30 T 5 ±40 T75 170 +.35 -0 150 160 +.30 - ±30 =F60 -90 +0) ~30 +0 155 +.20 =F 5 ±30 =F55 ±80 +75 ~~50 +.25 -0 160 T20 ±40 =F60 ±80 +80 ~60 +40 ~20 +0 165 ±60 T75 -90 p. - 60 > 130 t« -0 170. +.80 ~70 ^60 _50 t" ~30 +.20 10 +0 175 ^40 +.35 130 ^25 ~20 > Iio l> -0 180 + - + - + - + o - +0 Page 59 CHAPTER IV TREATMENT OF NON-SINUSOIDAL ALTERNATING QUANTITIES. THE USE OF THE OSCILLOGRAPH 43. General. — Following out the plan, as stated, of consider- ing some of the more important matters relating to special in- vestigation of transmission line phenomena, this chapter, which deals with the methods of obtaining the periodic curve and of utilizing the results of harmonic analysis of the same, is included. For a large part of alternating current investigation the complex quantity method of treatment, to which such prominence has been given by the work of Steinmetz, is the ideal one. A very brief outline of the method is sufficient here since detailed treatment is to be found in most all of the modern texts. The simplest alternating quantity may be described as one which varies harmonically with time between numerically equal positive and negative limits. Such a variation is represented algebraically by a trigonometric function — either cosine or sine, or both, depending upon the arbitrarily selected instant from which to count time. Thus, the displacement of a clock pendulum (whose motion is approximately harmonic) from its position of equilibrium would be represented by a cosine function of time if the zero of time was chosen as the moment when the pendulum was in an extreme position away from its equilibrium position; while if the zero of time was taken at the instant the pendulum passes through its equilibrium position, the displacement there- from is represented by a sine function of time. This applies equally to alternating electrical quantities, in that we may assign to one of the quantities in the system either a cosine or a sine (or a com- bination of both) variation by the proper selection of our arbitrary zero of time. All other quantities must be referred to this same zero, and thus such an arbitrary selection of the particular trigono- metric function for one of the quantities can be made for this one quantity only — all others being represented by such functions as will give to them their actual phase displacement from the 60 THE POLAR DIAGRAM 61 initial quantity. By properly selecting the zero of time important simplifications may often be made in the form of the analytic expressions for physical phenomena. 44. The Polar Diagram and Vector Representation of Alter- nating Quantities. — Let the alternating current flowing in an electrical circuit be represented by i = a cos cat. (1) Plotted in rectangular coordinates there results the familiar cosine curve. Plotted in polar coordinates with OX taken as the .- ^js Fig. 12. — The Circles Representing Trigonometric Functions Plotted in Polar Coordinates. OX is the Reference Line. reference line (Fig. 12) and with the angle 6 = <at as independent variable counted positive in the clockwise direction, the curve becomes a circle whose diameter lies along the reference line OX and has a length a. This arbitrary selection of positive direction of rotation is opposite to that usually used in mathematical dis- cussions. It is used thus in order to lead to a standardized form of expression for impedances. Suppose the current i to flow through a circuit containing 62 NON-SINUSOIDAL ALTERNATING QUANTITIES resistance, inductance, and capacity; r, L, and C, respectively. Then by the fundamental laws of the electric circuit, the impressed e.m.f. at any instant is e = ri + Lj t + -gfi dt, (2) which, for a current flow represented by equation (1), becomes e = ar cos ut — a(uL ^jsinut. (3) The e.m.f. thus consists of two components, a cosine and a sine component when referred to the same zero instant as the current i. The two dotted circles in Fig. 12 represent these two compo- nents of e, where e = ei + e 2 , ei = ar cos cat, fy = — a ( laL -p, J sin at. (4) The single curve representing e as the resultant or sum of the two circles, e x and e 2 , has its diameter displaced from the horizontal by an angle whose tangent is uL ^ Fig. 13 shows the two circles, e and i. For convenience, the circles may be omitted from the diagram and the diameters only, retained. Further, since, in general, effective or root-mean-square values are desired, the diameters may be drawn to such a scale as to represent these values which are —7= times the maximum. Such a representation of alternating v2 quantities constitutes the so-called "vector diagrams." They are in no sense vector diagrams, however, since the lines drawn therein are, in effect, only the diameters of the polar circles which have arbitrarily been omitted, and do not possess any of the properties of a physical "vector." The only point of resemblance lies in the fact that the parallelogram law of combination of such diameters holds true for different components of the same alternating quan- COMPLEX QUANTITY REPRESENTATION 63 tity. Since the term "vector diagram" has become so familiar and receives, such wide usage, it appears proper to retain it here, the distinction between such diagrams and the true vector diagram of mechanics being recognized, however. Fig. 13. — The Polar Circles Representing Voltage and Current. 45. Complex Quantity Representation. — Having reduced the representation of trigonometric functions to the simple straight lines of the vector diagram, further simplification is to be made. The positions of the ends of the vectors in the diagram may be specified in a system of rectangular coordinates, and the diameters of the polar circles which are represented by these vectors are completely located thereby. In naming the coordinates of the end of the vector, horizontal distances are taken positive to the right and negative to the left of the origin. Distances in a per- pendicular direction are positive when above and negative when below the horizontal axis. To distinguish these latter components from the horizontal, the prefix j is used. According to this convention, the two alternating quantities, i and e, as shown in Fig. 13 by their polar circles and in Fig. 14 by 64 NON-SINUSOIDAL ALTERNATING QUANTITIES the corresponding vectors, and whose equations in time are given by (1) and (3), may be most conveniently expressed by In (5), capital letters are used, indicating an alternating quantity specified in vector notation. Cosine functions are always repre- (5) Fig. 14. — The Vectors Representing the Diameters of the Polar Circles. sented by the term not affected with the symbol j. A sine wave of the form becomes fy = —a[ o)L 7: ) sin wt 27 2 =+ja(o,L-^) in the new notation. Negative sine functions take the +j as prefix, and positive sine functions the — j, while the cosine functions require no prefix and retain their original algebraic sign when expressed in the new notation. Alternating quantities are thus expressible in either of the two analytical forms — by trigonometric functions, or by means of the simplified notation above. The vector C = a + jb may THE COMPLEX OPERATOR 65 immediately be written c = a cos cot — b sin oit , unless effective values are represented by the vector notation, in which case, to give true instantaneous values, we must write c = V2 (a cos at — b sin cat). 46. The Complex Operator — Ratio between Two Alternating Quantities Expressed as Vectors. — The method of expression of alternating quantities by the use of so-called vectors, and the simplified notation for such, involving the use of the prefix j, does not imply that any significance be attached to the symbol j other than that of a mere distinguishing mark to designate a distance above or below the reference axis in the polar diagram. Great use is made, however, of a further assigned significance. In equation (5) the ratio between e.m.f. E and current I may be formed, E and I retaining their vector expression. ar + jatooL - — ^j I a + jO « serves as a defining equation for the quantity Z, which must have the same physical dimensions as electrical resistance, being the ratio of voltage to current. By division in (6) This quantity Z cannot represent any alternating quantity of a type similar to E or 7, for if the ratio of instantaneous values is taken at — a ( ojL -pz I sin ojt . ., . V wC/ = r - ( coL — J a cos oit \ oiC/ ar cos " tan oit 1 and this ratio passes recurrently through positive and negative infinite values, while writing (7) in a trigonometric form by the transformation defined in the preceding paragraph yields r cos oit — (oiL ~ ) sin oot t(— a which is a totally different result, and therefore wrong. The quantity Z as defined by (7) is merely an operator. Multi- plication of the current value I by this operator yields the vector 66 NON-SINUSOIDAL ALTERNATING QUANTITIES expression for e.m.f., or inversely, division of the voltage expres- sion E, in vector form, by this operator yields the expression for current, 7. To be such an operator, this quantity Z must be in- dependent of the arbitrary selection of the zero instant of time. This property may be made the basis for an interpretation of the significance of the symbol j. Take the origin of time so that the expression for current as given in equation (1) becomes i = a sin <at. Then e = ri + L Jt + lj idt = ar sin wt + a ( coL ~ J cos at, and in the vector notation l = 0-ja, from which the ratio aLL--^)-jra Z= | = _V «£/ (8) I -ja Equating the values of Z as given by (7) and (8), {" L -^c)- jr = r+j ( wL -^) 1 -J from which, after multiplying throughout by — j, -f = 1, or j = \/^I. (9) The assignment of this numerical significance to the quantity j does not prevent its use as a designating symbol for the upward direction in the vector diagram, while it does enable all formal algebraic operations, multiplication, addition, extraction of roots, etc., incident to computations relative to complicated electrical systems, involving the use of such operators as the complex quantity Z, or its reciprocal Y, to be rigorously carried out. The reader will find full discussion of the use of such complex quantity LIMITATIONS OF THE COMPLEX QUANTITY METHOD 67 representation of impedances, Z, in connection with the vector representation of alternating electrical quantities in any treatise on alternating currents — particularly in Steinmetz' "Alternating Current Phenomena," where the method is developed in great detail and applied to a large number of special cases. 47. Transformation from Trigonometric Functions of Time to Complex Representation. — The points for which emphasis is sought here are: 1st, the ready transformation from a trigono- metric expression for an alternating quantity, a cos cot -\- b sin cot, to a vector representation, a — jb, and vice versa; and 2nd, the fundamental difference between the vectors used in the diagram which represent alternating quantities, and the ratios between such quantities which are the complex operators and can in no sense be drawn in such a vector diagram, although they are repre- sented algebraically by a similar notation. Thus E = e\ + je 2 represents a definite alternating quantity, ei cos cot — e 2 sin cot, and as such may be drawn as a vector in the conventional vector diagram, ei and e 2 being the coordinates of the end of the vector, while Z = r + jx represents only the complex operator which signifies the ratio between voltage impressed upon and current flowing in a circuit of resistance r and inductive reactance x. For discussion of the quantities, resistance, reactance, conduct- ance, and susceptance, as well as the allied quantities, impedance and admittance, the reader is referred to any text on alternating currents. 48. Superposition. Use and Limitations of the Complex Quantity Method. — In any electrical circuit whose physical properties, resistance, self-inductance, capacity, etc., remain con- stant regardless of the magnitudes of the voltages or currents present, any number of impressed voltages, if they be of the same frequency and pure sinusoidal wave shape, will produce currents and potential differences among the several portions of the system, all of which are of the same frequency and of a pure sinusoidal form. Calculations of phenomena in such circuits are made using the vector notation for alternating quantities and the complex form for the operators which represent the properties of the various portions of the circuit (impedance, admittance, etc.), in exactly the same manner as would be done for a similar combination of 68 NON-SINUSOIDAL ALTERNATING QUANTITIES resistances only under the influence of unvarying, unidirectional impressed voltages. Kirchhoff's Laws, with impedance as the generalized term substituted for resistance, form the basis for the development of equations. If, in such a system in operation, a source of e.m.f. of different frequency be inserted, the calculations relative to the phenomena of the existing frequency still hold true, but in addition to these phenomena there is added a new set of the same frequency as the new e.m.f. whose frequency differs from the other. In other words, the method of superposition is to be employed in making calculations, and, in fact, the phenomena may themselves be re- garded as the resultant of several component parts. It must be remembered that the scheme of superposition is merely a con- ceptual one, however, and that at any instant there is but one actual current or potential difference present in a given portion of a system. It is necessary to realize that not only is the method of superposition, but also the use of vector representation inap- plicable, when the properties of a circuit vary with variations in the magnitude of currents or voltages. A case in point is that of an iron cored reactance, where the permeability of the core, and thereby the self-inductance of the circuit when defined in any way whatsoever, is not a constant, but varies with the magnetizing force. In such cases, sinusoidal impressed e.m.f.'s lead to non- sinusoidal currents, and vice versa. As opposed to the above limitations may be mentioned the case in which the apparent constants or properties of a circuit vary with the changes in frequency, but not with changes in magnitudes. Such cases may be rigorously treated by the complex quantity (vectors and complex operators) method provided care is used in selecting the values of the circuit properties which exist at the frequency under discussion. For example, if a circuit contains an air-cored inductance coil, it would be perfectly proper to base vector quantity calculations for this circuit on the values of effec- tive resistance and inductance of this coil at the frequency in question. In general the effective resistance and effective in- ductance of a compactly wound coil will change with change in frequency, but these changes arise from the fact that the coil to which we attach the notion of simple resistance and self-inductance really consists of a very intricate combination of inductances, resistances, and capacities (capacities between adjacent turns) and NON-SINUSOIDAL ALTERNATING QUANTITIES 69 that the conductor of which it is composed is of finite size and thus has an unequal current distribution over its cross section, so that from the very nature of the case we could not expect the apparent resistance and self-inductance to be the same at different fre- quencies. By regarding such a coil as an infinitely complicated system of simple parts of circuits we would still expect the proper- ties of the coil to be independent of the magnitude of current flow; and such is the case, the effective constants determined for any particular frequency and current strength holding true for all values of current at this same frequency. In such a case the only significance the terms "effective resistance" and "effective self- inductance" can have are respectively the real portion and - times the reactance, or j term, of the observed complex ratio WWV^WW •-AAAAA- >Vib. L =0.00140 henrys. r =0.200 ohms. C'=10xl0" 6 farads. r'=30.0ohms. Fig. 15. — Oscillograph Distortion Circuit for Current Measurements. between impressed voltage and current at the particular frequency used. 49. Numerical Example of Complex Quantity Treatment of Non-Sinusoidal Alternating Quantities. — To illustrate the method of treatment of non-sinusoidal periodic alternating currents by the vector method, application to some special cases will be shown. Suppose that in order to determine the wave-shape of the current flowing in a given conductor an oscillograph is used, connected as shown in Fig. 15. In this figure i is the current in the main conductor, whose wave shape is to be determined. The oscillograph galvanometer is connected in series with a resistance 70 NON-SINUSOIDAL ALTERNATING QUANTITIES r' and a condenser C", the whole being shunted around an induc- tance coil of resistance r and self-inductance L. The constants of the two circuits are adjusted so that the proper amount of cur- rent for the operation of the oscillograph galvanometer is diverted through the same. Reasons for selecting a circuit arrangement as shown will be given later. The upper curve in Fig. 16 shows the wave shape as recorded by the oscillogram. By means of a direct current calibration of the galvanometer, the amount of current per unit linear deflection of the spot of light from the zero-point may be determined, and thus the scale of ordinates may be such as to indicate directly the number of amperes of current flowing in the oscillograph vibrator circuit. The y coordinates of the upper curve in Fig. 16 give instantaneous values of the current i'. Assuming for the present that the oscillograph records accurately the instantaneous value of current flowing therein, correction must be made for the distorting effect of the circuit arrangement in order to determine the actual wave shape of the current in the main circuit, since it is only when non-inductive resistances are used, both for the shunt and for the vibrator circuit proper, that the wave shape as shown by the oscillogram will be the same as that of the line current. Analyzing the oscillogram for i ' by means of Analysis Forms F and G, from 18 ordinates measured over one-half wave, and with the origin taken as indicated in the figure, the equation of the vibrator current was determined to be i> = -0.0204 cos + 0.0347 sin +0.0210 cos 3 + 0.0018 sin 3 -0.0034 cos 5 - 0.0072 sin 5 -0.0038 cos 7 + 0.0037 sin 7 -0.0052 cos 9 - 0.0002 sin 9 -0.0003 cos 11 + 0.0058 sin 11 +0.0001 cos 13 - 0.0003 sin 13 +0.0004 cos 15 - 0.0003 sin 15 , - 0.0004 cos 17 - 0.0005 sin 17 0. (10) The fundamental frequency is 60 cycles per second. Then, co„ = 2 irn (60) = 377 n = angular velocity for nth harmonic. If Z„ = r + joj n L = impedance of shunt, NON-SINUSOIDAL ALTERNATING QUANTITIES 71 and Z n ' = r' — j — p^ = impedance of vibrator circuit, by the application of Kirchhoff's Law it is easily shown that in vector notation 1 + §Ai (ID The vector expressions for vibrator current may be written immediately from equation (10) ; thus, for example, using maximum values, W = 0.0210 -j 0.0018, while the term in parentheses in (11) is to be computed from the values of the circuit constants. The following table indicates the main features and results of the calculation by equation (11). TABLE IX Correction for Oscillograph Distortion Circuit (Maximum values of current) n z n ' Z n (' + fc') /»' In 1 30-/265 0.20 +/0. 528 -419 -/215 -0.0204-/0.0347 1.10+/18.9 3 30-/88.3 0.20+/ 1.58 -51.8-/25.6 0.0210-/0.0018 -1.13-/0.44 5 30-/53.0 0.20+/2.64 -18.1-/12.8 -0.0034+/ 0.0072 0.154-/0.087 7 30-/37.9 0.20+/3.70 -8.76-/8.64 -0.0038-/0.0037 0.001 +/0.065 9 30-/29.4 0.20+/4.75 -4.93-/6.55 -0.0052+/ 0.0002 0.027 +/0. 033 11 30-/24.1 0.20+/5.81 -2.97-/5.30 -0.0003-/0.0058 -0.030 +/0. 019 13 30-/20.4 0.20+/6.86 -1.84-/4.46 0.0001 +/0. 0003 0.001-/0.001 15 30-/17.7 0.20+/7.92 -1.14-/3.84 0.0004 +/0. 0003 0.001-/0.002 17 30-/15.6 0.20+/8.98 -0.66-/3.37 -0.0004+/ 0.0005 0.002 +/0. 001 The complex values of I n give the following equation which represents the line current i as a function of time when referred to the same zero as the oscillograph current i' . i= 1.10 cos 6 - 18.9 sin 9 -1.13cos30 + O.44sin30 +0.154 cos 5 + 0.087 sin 5 +0.001 cos 7 - 0.065 sin 7 +0.027 cos 9 - 0.033 sin 9 -0.030 cos 11 - 0.019 sin 11 +0.001 cos 13 + 0.001 sin 13 +0.001 cos 15 + 0.002 sin 15 +0.002 cos 17 - 0.001 sin 17 0. (12) 72 THE USE OF THE OSCILLOGRAPH -24 Fig. 16. — The Distortion Produced by the Circuit of Fig. 15. Upper: Oscil- lograph Vibrator Current, i'. Lower: Main Line Current, i, by Analysis. The lower curve in Fig. 16 shows the wave represented by equation (12), in the proper phase position with reference to the upper curve which represents the oscillograph vibrator current. DISTORTING CIRCUITS FOR USE WITH OSCILLOGRAPH 73 A comparison of the two curves, i and i', should immediately show the advantage of using a circuit connection as illustrated by- Fig. 15. The higher harmonics are very much accentuated in the vibrator circuit, and thus in the resulting oscillogram, thereby rendering their determination much more accurate than would be possible if an undistorted wave shape were used. The amount of the distortion is strikingly illustrated by a comparison of the Z' different frequency values of the quantity 1 + ■=-, as given in Zi Table IX, and which represent the ratios of line current to oscillo- graph current. 50. Distorting Circuits for Use with the Oscillograph. — The oscillograph is used to obtain one of two things — either the vave shape of the current in some conductor, or the wave shape of a potential difference or an electromotive force. These, at any rate, will be the quantities of particular interest in a consideration of transmission phenomena. With the exception of an electrostatic form of oscillograph, all types of oscillographs require for their operation a current of appreciable magnitude, since they are based upon either an electromagnetic or hot-wire principle, and thus they cannot indicate potential differences directly but only through the medium of a current produced thereby. One of the most popular commercial oscillographs requires a current of approxi- mately 50 milliamperes to operate it satisfactorily, so that in many instances the effect of a current flow of this magnitude on the phenomena in the system under investigation cannot be neglected. For the recording of potential differences, the electro- static form of instrument is theoretically the ideal one — its electrostatic capacity being so small that the current consumed thereby is negligible, except at frequencies far higher than the vibrating system could record. The electromagnetic type, on the other hand, affords almost an ideal instrument for the indica- tion of current, since the resistance of the galvanometer circuit may be made as low as one or two ohms, and which, for the indication of currents of large magnitude, could be connected around a shunt of very low resistance. The whole combination would thus cause a drop in potential of only one-twentieth to one-tenth of a volt, a quantity usually negligible. In a large number of instances where it is definitely known that the effect of the oscillograph circuit in altering the ex- 74 THE USE OF THE OSCILLOGRAPH isting phenomena in the system can be neglected, or when it is desired to record the phenomena which occur as affected by the oscillograph regardless of whether or not they are the same as would occur with the oscillograph removed, a very great increase in the accuracy of determination of the various' frequency com- ponents can be secured by so arranging the oscillograph circuit that all of the harmonics whose values are desired produce cur- rents through the galvanometer of the same order of magnitude. In the numerical example just considered this condition was secured by means of the circuit combination shown in Fig. 15. With such a circuit arrangement any particular harmonic may be accentuated in the oscillogram, for by adjusting the values of L and C a condition approximating current resonance for this particular frequency may be secured, in which case the vibrator current may be made much greater, even, than the total line current. In order to avoid difficulties from current resonance where such is not desired, it is always well to insert considerable resistance in series with the galvanometer and condenser, and while observing the wave shape on the visual screen to reduce this resistance until the desired distortion is obtained. If one fre- quency predominates to an undesirable extent, a change in the condenser capacity will usually remedy the difficulty. In using any oscillograph circuit containing capacity, it is well to always make the first connection to the circuit under test through a comparatively high resistance which subsequently is gradually removed. This prevents a transient flow of current, into the condenser, of a magnitude sufficient to burn out the protective fuses placed in series with the galvanometer. In this discussion of distorting circuits to be used for accentuat- ing the higher harmonics, it is of course to be understood that the scheme is to be applied only when the system is operating in a steady condition. It is obvious that an investigation of transient phenomena would not permit of the use of circuits of the type discussed here. If the extreme distortion possible with the connection shown in Fig. 15 is not desired, a very convenient connection consists in shunting an inductance coil with a non-inductive vibrator circuit — that is, using the connection in Fig. 15 with the condenser C short-circuited. Another connection which would yield approxi- mately the same result would be that in which the galvanometer DISTORTING CIRCUITS FOR USE WITH OSCILLOGRAPH 75 circuit containing a condenser and resistance was shunted around a non-inductive resistance. In making an oscillogram of the current flowing in a high voltage system, where it would be impracticable to utilize any- direct shunt connection on account of the large potential differ- ences, most satisfactory results may be obtained by using a mutual inductance coupling between the high voltage circuit and the oscillograph circuit.* For this purpose, air-core inductance coils of large diameters and small winding sections will permit of a fairly close electromagnetic coupling with a separation between coils vvwWIif M mu\ l-AAAAA iVib. Fig. 17. — Electromagnetically Coupled Current-distorting Circuit. great enough to eliminate any danger from the high potential system. One of the coils is connected in series with the high voltage circuit and the other is connected in series with the oscillo- graph vibrator and the proper regulating resistance. If great distortion is desired, the oscillograph circuit may contain a con- denser. For a circuit connection as shown in Fig. 17 the following equation expresses the relation between oscillograph and line cur- rents, from which the results may be corrected so as to yield the true wave shape of line current. n — Jl/f -* n • (13) * The use of current transformers for this purpose is not recommended if a distortion circuit is to be employed, for the impedance of such a circuit may require a voltage sufficient to prevent a proper current transformation. 76 THE USE OF THE OSCILLOGRAPH Equation (13) is to be used in exactly the same manner as equation (11), illustration of which has been given. If the con- denser is not present, that is, equivalent to being short-circuited, the term l/w n C" is zero, in (13). If the wave shape only of the line current is desired, and not the absolute magnitude, it is obvious from (13) that the value of the mutual inductance, M, need not be known in order to make the reduction. This is a distinct advantage, and in fact for almost any investigation of current wave shape, the form of circuit shown in Fig. 17 is superior to that shown in Fig. 15, since a flexibility of control by variation of the mutual inductance is possible; and further, the labor of computation necessary for reducing the oscillographic data is considerably less by equation (13) than by equation (11). Of course care must be exercised to prevent extraneous varying magnetic fields from affecting the oscillograph circuit, for in such an event, the determination of the line current would be vitiated. 51. Oscillographic Determination of Wave Shape of E.M.F. — To secure oscillograms representing wave shapes of voltage by means of either the electromagnetic or hot-wire type of instru- ment, the current flow must be limited by some form of series impedance. If non-inductive resistance is used, the wave shape of the current through the vibrator will be identical with that of the impressed voltage at the time of recording, though on account of the current consumed this voltage wave shape may differ from that which would exist on removal of the instrument. If dis- tortion of wave shape is desired, the current flow through the galvanometer may be limited by inserting series capacity, and at the same time decreasing the resistance. If the resistance present be small in comparison with the impedance of the condenser, the distortion secured by this scheme of connection will be approxi- mately proportional to the order of the harmonic. This is ap- parent from the fact that the impedance of a condenser changes in inverse proportion to change in frequency. Very much greater distortion may be secured by the use of a circuit of the type shown in Fig. 18. In this diagram, C and r constitute the chief current limiting impedance, while the parallel circuits, C, r' and L", r", are for the purpose of securing a selective current flow through the galvanometer, and may be of much lower impedance than the circuit consisting of C and r. The equation DETERMINATION OF WAVE SHAPE OF E.M.F. 77 necessary for the determination of the line voltage E from the current I' as represented by the oscillogram is E n = [Z n + Z n ' + " jj \l n ' (14) where Z„, Z„', and Z n " are the impedances for the nth harmonic of the respective portions of the circuit. r" L" ^AAA-^nnfFn L-AAA/^ Fig. 18. — Distortion Circuit for Determining Voltage Wave Shapes. The current consumed by the entire circuit arrangement is I n = (l+^pjl n '. (15) To illustrate the use of such a circuit the following numerical example is considered. In this case the vibrator current wave shape is an assumed one, the values of the various frequency components having been selected at random, but with care that the total vibrator current be of the proper magnitude. Even harmonics are included. 78 THE USE OF THE OSCILLOGRAPH The equation for the vibrator current i' is i' = 0.0158 cos - 0.0054 sin -0.0060 cos 2 + 0.0058 sin 2 -0.0101 cos 3 + 0.0054 sin 3 +0.0014 cos 4 + 0.0015 sin 4 - 0.0026 cos 5 - 0.01 12 sin 5 +0.0004 cos 6 - 0.0064 sin 6 +0.0079 cos 7 + 0.0022 sin 7 0, (16) and the wave shape represented thereby is shown by the upper curve of Fig. 19. Let the circuit constants shown in Fig. 18 have the following numerical values. r = 200.0 ohms. r' = 12.0 ohms, r" = 10.0 ohms. C = 0.50 X 10-* farads. C" = 9.0 X 10" 6 farads. L" = 0.150 henrys. Table X shows the principal numerical values entering into the transformation from the observed harmonics in the oscillogram, , that is in i', to those of voltage e. TABLE X Determination of Voltage Wave Shape from Oscillograph Vibrator Current. Connections as per Fig. 18 (Frequency of fundamental, 60 cycles) n 1 2 3 4 5 6 7 Z n 200 -J5305 200 -J2652 200 -j 1768 200 -j 1326 200 -/ 1061 200 -^884 200 -J758 Z n ' 12.0 -j 294.7 12.0 -j" 147.4 12.0 -j'98.2 12.0 -j 73.7 12.0 -.7*58.9 12.0 -.7*49.1 12.0 -J42.1 Z n " 10.0 +;'56.5 10.0 +?113 10.0 +7*169 10.0 +7*226 10.0 +7*282 10.0 +7*339 10.0 +7*396 En In -6620 +j 20,900 -630 +7 574 -87 -j 876 +58 -j 986 +118 -.7*912 + 148 -.7*814 + 166 -j 727 In 0.0158 +7 0.0054 -0.0060 -jO.0058 -0.0101 -.70.0054 0.0014 -.7*0.0015 -0.0026 +7*0.0112 0.0004 +/ 0.0064 0.0079 -./0.0022 En -218 +/294 7.11 +7*0.22 -3.86 +7*9.32 -1.40 -j 1.47 9.91 +7*3.69 5.28 +7*0.62 -0.29 -j6.il (Slide rule accuracy in computations.) DETERMINATION OF WAVE SHAPE OF E.M.F. 79 Fig. 19. — The Distortion Produced by the Circuit of Fig. 18. Upper: Oscil- lograph Vibrator Current, i'. Lower: Main Line Voltage, e, by Analysis. 80 THE USE OF THE OSCILLOGRAPH The equation for instantaneous values of e is then, from Table X, e = -218 cosd- 294 sine +7.11 cos 20- 0.22 sin 20 - 3.86 cos 3 6 - 9.32 sin 3 6 - 1.40 cos 4 6 + 1.47 sin 4 6 +9.91 cos 5 6 - 3.69 sin 5 6 +5.28 cos 6 6 - 0.62 sin 6 6 -O.29cos70 + 6.11sin7 0. (17) The lower curve in Fig. 19 was plotted point by point from equation (17), and serves very well to illustrate the magnifying effect of the circuit used on the higher harmonics. It should be noticed that for the second and third harmonics, the distortion is particularly great on account of the approximation of these frequencies to the resonant frequency of the circuits Z' and Z". This is indicated by the low values of the ratio y in Table X. All of the different frequency components of higher order than the first are magnified to about thirty times their normal values in the oscillogram. Using a smaller condenser capacity C", or a smaller self-inductance L", would prevent the great relative magnification of the second and third harmonics, but would still magnify those of higher order to a sufficient extent. CHAPTER V EQUATIONS OF MOTION OF THE GALVANOMETER MIRROR IN THE ELECTROMAGNETIC TYPE OF OSCILLOGRAPH 52. General. — In this chapter the equations of motion of a mirror, attached to the two supporting strips which constitute the galvanometer coil of the electromagnetic type of oscillograph are developed, with a view to showing the extent to which the indica- tions of such a device are in error when used to record the wave forms of uniformly alternating currents. The discussion also affords a good illustration of the use of the complex quantity notation as applied to alternating quantities other than currents and electromotive forces. In using the oscillograph to record continuously alternating currents, there is probability of two kinds of error, viz., change in calibration constant of the vibrator for different frequencies, and a time-phase displacement of the mirror deflection from the current flowing in the supporting strips. Departure of the damping force acting upon the mirror and strip from strict proportionality with the velocity of these parts would cause the mirror motion to be other than a pure sine wave even though the deflecting force be such. Because of the low velocity of these two parts, however, it is improbable that such an effect could be detected. In calculations relative to the mirror motion, the damping force will be assumed proportional to the velocity. If the moving system is not damped at all, the deflections of the mirror will at all times be in phase with the deflecting force for frequencies below the natural or free frequency of vibration, while for frequencies above this value, the displacement will reverse in phase. On account of the damping, the deflections will not be in exact phase coincidence with the force acting, but will, for fre- quencies below the free vibration frequency, be lagging with respect thereto. 81 82 EQUATIONS OF MOTION The inertia of the moving parts tends to cause a change in the calibration constant of the vibrator — particularly for frequencies approaching the natural frequency. At very high frequencies it is conceivable that the motion of the vibrator strips in the intense magnetic field produces an induced electromotive force of a magnitude comparable with that impressed upon the vibrator circuit. This matter will be referred to again. 53. Fundamental Equations of Motion of the Vibrator Strips. — Assume the entire length of the vibrator strip to be acted upon by a force a cos ut per unit of length, by virtue of a current i cos (A flowing in the strip. Let T = tension in the strip, in dynes, and m = mass of the strip per unit length, in grams. The fundamental differential equation of motion when the strip is immersed in a viscous liquid is d^s ds S^s T te>- k M + aC0S03t = m W (1) where' x is measured along the equilibrium position of the strip, and s is measured perpendicularly thereto, and thus represents the displacement, k represents the damping force per unit length of strip when the strip is moving with unit velocity. The solution of the above partial differential equation will yield a combination of decreasing exponential functions (decreasing with time) and trigonometric functions, so that under steady con- ditions after the transient terms have become zero the motion of the strip at any point in its length will be represented by the trigonometric terms which are not affected by the decreasing exponentials. Thus s = A cos ut + B sin ut, (2) where the integration constants A and B are functions of x and the physical properties of the system. Since s is a simple harmonically varying function it may be DETERMINATION OF INTEGRATION CONSTANTS 83 represented by the conventional complex quantity notation, and the variable t thus eliminated. Thus, for any arbitrarily selected origin of time, F = deflecting force = /i + .7/2, S = displacement of strip at any point = Si + j«2, V = -jt = jcoS = velocity, A = —rr = -j77 = — oPS = acceleration. at at 2 (3) Equation (1) becomes for the permanent condition of operation T~-kV + F = mA, T 6 ^- jkwS + F = -mco 2 ^. Combining terms and placing „ _ mat 2 — jku) (4) T N = F T' equation (4) reduces to ■N, the solution for which is S =Aic** x + A 2 e-i ax — N (5) (6) (7) in which Ai and A 2 are arbitrary integration constants which must be determined from known boundary or initial conditions. Since a 2 is itself complex, the quantity ja is complex, and therefore the exponentials become combinations of trigonometric functions and exponentials with real exponents. 54. Determination of Integration Constants. Their De- pendence upon the Moment of Inertia of the Mirror. — Let 2 1 = total length of vibrator strip, or I = distance from support at end to mirror at the mid-point of the strips. (8) 84 EQUATIONS OF MOTION At x = l f the slope of the strip at any instant must be such as to give to the point of contact with the mirror the same acceleration as (7) would yield for x — I; this on the assumption that the mirror causes no damping over and above that due to the vibrator strips, and further that the length of the mirror along the strip is negligible as well as any forces due to flexure of the strip at the point of contact. These conditions are not realized exactly, but to take them into consideration would hopelessly complicate the problem. It is not to be expected that the neglect of these matters will vitiate the solution to any great extent. Let 4 7 = polar moment of inertia of the mirror about a gravity axis parallel to the strips. 2 d = spacing of the two supporting strips, center to center. (9) The normal acceleration at the point of contact is d times the angular acceleration of the mirror, and the torque exerted by each half-strip is -Td d /\ dx} x= i The algebraic sign used in this expression for torque applies to the first half-strip only, but as all of the four half-strips exert equal accelerating forces upon the mirror, each may be thought of as accelerating one-fourth of the mirror, and our investigation confined thereby to one half-strip only. d?s dt 2 I Td? ds\ \ z= i I dx\ z - (10) d 2 s In the complex quantity notation,-^ = — &>*$. Let Td? H = —j- , and then at x = l f -tfS = - H — • — = S~- dx ' dx H (11) Equation (11) serves as one boundary condition for the deter- mination of the integration constants, in that it expresses the THE MIRROR MOTION 85 value of the first derivative of the function in terms of the function itself, at a given value of x. As a second condition, since there are two constants to be determined, S = at x = 0. Thus N A, + A 2 - - 2 = 0. or Substituting (7) in (11), and then placing x = I, jj Ai<t** + jj A*T** - ^ = jaAie** 1 - jaA*-** 1 , (12) or and At + A 2 Solving the two equations of (13) for A\ and A 2 , co 2 /co 2 , . \ . . (13) Ai» A 2 = a 2 /co 2 . \ . . /co 2 , . \ . . N (| 2 - ia )^-| 2 a 2 /co 2 . \ . . /co 2 , . \ ' \H ~ Ja ) ~\H + Ja ) (14) 65. The Mirror Motion. — If these values of A\ and A 2 be substituted in (7), the equation of motion for the strip at any point will be obtained. Since it is the motion of the mirror only that is of interest at present, the introduction of (14) in (7) and the substitution of I for x in the resulting equation will give the desired solution. S, as given by (7), used with the subscript m designates the mirror motion so obtained. 9 -E or _ ( e *w _ e -*i) _ 2 jo ti - 1. _ ( e *w _ (-M) — j a (ei al + e - ^) ti (15) 80 EQUATIONS OF MOTION Writing (15) in a trigonometric form, that is, substituting sines and cosines for the imaginary exponentials, S m = X 77 sin al — a H II sin al — a cos al which after simplification becomes S m — — a cos al — 1 // sin al — a cos al (16) which may be taken as the final form. Since a 2 is complex, a is also complex, and may be written a = u + jv. The square root is most easily formed by expressing a 2 as a scalar with its angle, and then taking for the scalar value of a the square root of the scalar value of a 2 , and for the angle of a, one-half the angle of a 2 . Thus, if a 2 = a+jb = c/2ji, where c 2 = a 2 + & 2 , and tan 2 = - , a we have a = Vc/fS — Vc cos /3 + j v'c sin j8. The trigonometric functions of the complex, a, are sin (w + jv) = cosh v sin u + j sinh v cos u, cos (u + jy) = cosh v cos u — j sinh v sin w. (17) Further reduction of (16) by introducing (17) is not desirable, except numerically, since the resulting expressions become too involved. 56. Application of Analysis to a Particular Case. — To illustrate the application of the foregoing development and to show the magnitude of the errors which may be expected in practice, we will consider the motion of the mirror in a vibrator used by the General Electric Company in their commercial form APPLICATION OF ANALYSIS TO A PARTICULAR CASE 87 of oscillograph. The dimensions of the moving parts are, approxi- mately: Silver alloy strips, 0.0070" X 0.00075". Total length of strips, 0.438". Tension per strip, 3.0 oz. Spacing of strips, 0.012". Glass mirror (thickness increased by 25 per cent to allow for cement used in attaching to strips), 0.060" X 0.017" X 0.0075" thick. Using 10.0 as the specific weight for the silver strips, and 3.0 for the glass mirror, the following values in C.G.S. units are ob- tained for the various constants. I = 0.555 cm. m = 0.340 X 10 -3 grams per cm. length. T = 83 ; 500 dynes. I = 1.75 X 10~ 8 gram-cm 2 . d = 0.01525 cm. The quantity k is not known, and its calculation from the known dimensions of the system and the viscosity of the liquid would be very difficult, but if the assumption is made that the vibrator strips are critically damped for the fundamental frequency of free vibration, its value may be determined. In Byerly's "Fourier's Series and Spherical Harmonics," the motion of a string of finite length vibrating in a resisting medium is discussed, and in his solution (on pages 113-115) we find, using the notation adopted here, the angular velocity of free vibration of fundamental frequency to be 1,/TV 2 fc 2 . so that the free vibration frequency when undamped would be the familiar equation for vibration of such a type. For critical damping, by (18), (20) 88 EQUATIONS OF MOTION For the numerical case in hand, ki = 30.16 and / = 7050. These values are, of course, not exact, since the mass of the mirror has been neglected, but they serve as a basis for an estimate as to the value of k. The free frequency of vibration as given by the manufacturers' data is 5000. Using the above tabulated numerical constants, three separate calculations for various frequencies have been made, taking three different values for A;, viz., fc = 0, k = 0.75 h = 22.62, k = 1.50 fci = 45.24, the results of which are given in the following table. The quan- tity -rp, as given in the table is proportional to the deflection of the mirror when unit current is flowing in the strips, and thus represents the calibration constant of the galvanometer. The angle <f> is the lag in time-phase position of the mirror deflection behind that of the current. TABLE XI Motion op Oscillograph Vibrator i = * = 0.75t, *= 1.50*, / s m <t> s« <t> s n 4> N deg. N deg. N deg. 0.1540 0.1540 0.1540 100 0.1541 0.1540 1.24 0.1539 2.48 500 0.1556 0.1547 6.29 0.1521 12.45 1000 0.1605 0.1567 12.78 0.1463 24.53 1500 0.1693 0.1599 19.66 0.1389 35.80 2000 0.1834 0.1641 27.16 0.1297 46.27 2500 0.2050 0.1687 35.45 0.1201 55.82 Figure 20 shows the curves plotted from the above data. The results of this numerical analysis are very interesting, and substantiate the statements made in the opening of this discus- sion, section 52. In no case can the oscillograph depict the true wave shape of a complex wave of current in the vibrator strips. If the damping is zero, the deflections are in their proper time- APPLICATION OF ANALYSIS TO A PARTICULAR CASE 89 phase position, but the calibration constant differs for the different frequencies — the higher frequency components being magnified in the oscillogram. For a damping a little greater than 0.75 hi, S the calibration constant of the vibrator, -—, will be practically unvarying with change of frequency, but in any case with damping N 1 v I. II. Zero Ijamping, 1 0.75 Critical Damping. 0.20 Deg III. 1.! OCr itica IDa mpi i?. 0.10 1/ 0.18 64 0.17 56 U, 0.16 48 S»i ^111 ( 0.15 40 0.14 32 III 'n 0.13 24 s 0.12 16 f 0.11 8 0.10 400 800 1200 1600 Erequency, /. 2000 2400 Fig. 20. — Characteristics of Oscillograph Vibrator Motion. Calibration Constant and Angle of Lag from Equation (16). present, an error in time-phase position occurs. The use of dis- torting circuits for the purpose of accentuating the higher har- monics in no way increases the percentage error introduced by the oscillograph, for such error is dependent upon the frequency only, and not upon the magnitude of the deflection (of course within the limits of permissible vibrator current). 90 EQUATIONS OF MOTION 67. The Intensity of the Magnetic Field Surrounding the Strips. — From the observed calibration constant of the instru- ct ment and the computed value of -^ it is possible to determine the flux density of the magnetic field in which the vibrator is placed. Let L = distance from mirror to observing screen. c = observed deflection of light spot per absolute unit of current in the galvanometer strips. / = current in vibrator strips, abamperes. Then D = cl = deflection of light spot on screen. (21) For the small deflections used, y-j = yj = -r 5 = angular rotation of mirror. (22) From equation (22), 2L at F By equation (5), however, iV = 7=, so that in (23) F = force per unit length = ^- = BI, (24) where B = flux density surrounding the vibrator strips. Thus -!?©■ » For the vibrator in question, the calibration constant was given as 0.0060 amperes per mm. deflection, from which c = 167 cm. per abampere of current. Also, L = 46 cm., approximately. o At zero frequency, from Table XI, the numerical value of -r? is 0.1540. Substituting these numerical values in equation (25), B = 15,010 lines per sq. cm., approximately. ELECTROMOTIVE FORCE INDUCED IN VIBRATOR 91 58. Electromotive Force Induced in Vibrator Strips. — The area swept over by a vibrator strip during a half-cycle may be determined approximately by assuming the strips to coincide with straight lines at the instant of maximum mirror deflection. From the known value of flux density, the total change of flux within the loop may be determined, and thus the induced e.m.f. Suppose the vibrator to be carrying a current of 0.060 amp. and thus giving a deflection of 1.0 cm. on the screen. By equation (22), then, S m = =-=r = 0.000166 cm., and the total area enclosed Z Li by the two strips when each is displaced by this amount at the center is 0.000184 sq. cm. The flux enclosed by this loop is 3W. = 0.000184 B = 2.76 lines. The effective value of induced e.m.f. is 4.44 $ max . / 10~ 8 volts. E = 12.3/ 10" 8 volts. Even for / = 5000 cycles per second, this gives only 0.00061 volts — an amount scarcely comparable with the minimum voltage impressed on the vibrator circuit. The resistance of the vibrator circuit may be reduced to approximately one ohm, thus requiring an e.m.f. of 0.060 volts to produce the amount of current under consideration. Thus under the most unfavorable conditions — highest frequency and lowest vibrator circuit resistance — the voltage induced in the vibrator strips by virtue of their motion in the magnetic field is only one per cent of the impressed, so that in no one case need any appreciable error be expected to arise from the cause here considered. 69. The Free Vibration Frequency of the System when the Inertia of the Mirror is Considered. — The vibrator mirror may oscillate freely at an infinite number of frequencies if the damping be not too great, and if the damping constant be zero these fre- quencies may be determined from equation (16) by equating the denominator in the parentheses to zero and then finding by trial the values of a> which satisfy the equation. This procedure amounts to finding the values of oo for which S m will have a finite value even though there be no current flowing in the vibrator strips — that is, N = 0. Thus, for free oscillation, undamped, jj sin al — a cos al = 0. (26) 92 EQUATIONS OF MOTION For k = 0, by equation (5), , mo) 2 . fm « = -f~> or a = w Vr' so that by replacing H by its equivalent, equation (11), ^sinV^-v/^cosyffco-O, from which, by dividing, 4 /m 7 VmTd? tan Vr^ = -^— ' tan" «/ (27) (28) either of which expressions may be used to determine w. A graphical method applied to the first of equations (28) yields sufficiently accurate results with very little labor. The two curves, y' = tany pz« and V = — f — » Id) may be plotted, using co as independent variable, and their inter- sections noted. The values of w at these intersections are those corresponding to free oscillations of the mirror. Since the second curve, y", is asymptotic to the axis of «, the intersections of the two curves give values of w more and more nearly equal to those for which tan J%U-0, or --«rv/^' where n is any integer. If no mirror be present the quantity I becomes zero in the equations, and then (28) reduces to or o)\ = tV/ — tan-^oo), _|/=o l ▼ m (29) THE FREE VIBRATION FREQUENCY OF THE SYSTEM 93 where n is any integer. For n = 1, the fundamental frequency of vibration for the strips alone becomes 'L-£L-n^' (30) a result identical with that given in equation (19). 5 4 V 3 r 2 a).. 1 (O y" 6 )x 1 o- 1 I 1 8 a i | 5 G 7 9-*" B 10 -1 v/ -2 Fig. 21. — Determination of the Free Frequencies of Vibration of Oscillograph Mirror, wi = 32,300. « 2 - 102,500. For the vibrator and mirror under consideration, equation (28) gives the numerical result, 70,808 tan (0.000035415 a) = (3D Fig. 21 shows the two curves, 70 808 y' = tan (0.000035415 co) and y" = — J— , from the intersections of which the first two free vibration fre- quencies are 3|300 102,500 _ Ztt ait 94 EQUATIONS OF MOTION The first of these values, 5150, may be compared with the free vibration frequency for the strips alone, 7050 cycles per second, as given by equation (19). The addition of the mirror to the system reduces the free vibration frequency by 1900 cycles per second. It is to be noticed that equation (29) gives no free vibration frequency which is an even multiple of the fundamental frequency, obtained by placing n = 1, even though it is known that a stretched string or strip may oscillate at any frequency which is a multiple of the fundamental. It is to be remembered in this connection that equation (16), on which (29) is based, refers to the motion of the mid-point of the stretched strip only, and that for even multiples of the fundamental frequency this point would be at a node and therefore have no motion. For all odd multiples the mid-point lies at an anti-node, and thus has a motion — the result shown by (29). CHAPTER VI ALTERNATING CURRENT PHENOMENA IN AN IDEAL LINE 60. The Ideal Line. — By an ideal line, from the point of view taken in the preparation of this chapter, is meant one which is characterized by four electrical properties per unit of length, viz., line resistance, inductance, conductance, and electrostatic capac- ity. The line is supposed to be uniform throughout, so the values of these four constants or properties are constant for every unit of length. These constants must be regarded as the effective values of such, since the apparent properties of a transmission system carrying alternating currents are dependent upon the fre- quency of the line voltages and currents. A discussion of such changes will follow, for here it is sufficient to consider the constants as effective values and to develop the equations for the alternating phenomena in terms of them. 61. Fundamental Circuit Constants. — In general, for sym- metrically arranged polyphase systems it is desirable to take the properties of each conductor as referred to the neutral of the system, although in the case of single phase transmission with two conductors it is satisfactory to consider the conductor with its return in determining the physical constants. In the first case, voltages are counted for each conductor to neutral, while in the second case, the voltage between wires is used. Let r = line resistance per unit of length, ohms. g = line conductance per unit of length, mhos. L = self-inductance per unit of length, henrys. C = electrostatic capacity per unit of length, farads. Since the numerical values of the line constants do not change (on the assumption of an ideal line) for a fixed frequency, a sine wave of voltage impressed on the line will produce only sine waves of voltage and current throughout as long as the load is not of such 95 96 ALTERNATING CURRENT PHENOMENA a character as to cause a distortion of wave shape. This is in accordance with the discussion in section 48, Chapter IV, and permits of a representation of the line constants at this fixed frequency by means of the conventional complex quantity nota- tion, impedance and admittance. Z = r -\-j2 vfL = impedance per unit length. Y = g + j 2 x/C = admittance per unit length. x = 2vfL, z=Vj* + 6 = 2ir/C, y=V^T6" 2 (1) In the two quantities, Z = r -\- jx and Y = g + jb, it is to be noted that the "j" terms are both of the same algebraic sign — a matter over which students very often become confused, since an admittance, Y, as the reciprocal of an impedance, Z, reverses the algebraic sign of the "j, " or imaginary, term. In this instance, however, the admittance Y is not the reciprocal of, nor in any way related to, the impedance Z — the former being the conductance and capacity admittance from wire to neutral, or between wires, while the latter is the inductive impedance of the wire, or wires. 62. Fundamental Differential Equations and Solutions. — In the alternating current system the electrostatic capacity is always present, and therefore the admittance Y, so that in no case can it be said that the current is of uniform value throughout the line length — a condition approximately realized in the case of direct current transmission over a line with good insulation. For this reason the exact solution for the alternating current problem will be given first, and approximate solutions discussed afterwards, with the rigorous expressions on hand for comparison. The fundamental differential equations for current and voltage along the line are set up in exactly the same way as for the direct current problem, equation (1), Chapter II, using Z and Y in place of r and g, respectively, f-™;f-*x, from which drl §-ZYI = 0, ^-ZYE=0. (2) AUXILIARY CIRCUIT CONSTANTS 97 Equation (2) is exactly similar to equation (3), Chapter II, the constant ZY taking the place of rg (to which ZY reduces for zero frequency), so that the complete solution may be written immedi- ately, following equation (9), Chapter II. e = \ [{e + /„ y/f) t *ra + (e - h y/f) •-*** ] , (3) where E and 7o are the voltage and current respectively at the load and E and I are the voltage and current respectively at a distance I from the load end. All of these four quantities are in general complex, since they represent electromotive forces or currents which are alternating. 63. Auxiliary Circuit Constants. — Since both Z and Y are complex, the two quantities \ y and VZY are, in general, both complex. Let V=VZY = a+jl3 (4) serve as a defining equation for V, a, and /3, which thereby become constants per unit of length of the transmission system, a and /3 may be explicitly expressed in terms of the fundamental line con- stants by the following method : (a+i/3) 2 = (r+jx)(g+jb), or a 2 + 2 ja& — /3 2 = rg — xb + jgx + jrb, from which by equating reals and imaginaries, a 2 — /3 2 = rg — xb, (4a) 2 aP = gx -f- rb. J From these two equations, as Steinmetz shows (Transient Electric Phenomena and Oscillations), « - V| {zy -xb + rg),) . g . = Vi (21/ + xb - rg)\ Though convenient for some purposes it is not desirable to use equation (5) for the calculation of numerical values of a and /3, Darticularly when the line resistance r and leakage g are small in 98 ALTERNATING CURRENT PHENOMENA comparison with x and b, respectively. In such an event the product zy differs very little from xb, and in the expression for a the resulting difference, zy — xb } would be difficult to obtain with accuracy. Besides, in any case the numerical work required is excessive. It is better to form Z m r+jx = z/0f Y = g+jb = yld JI VZY m « + j0 - V - Vzy /h (0. + B y ) = vl$, a = v cos 6 V , = v sin 8 V , and at the same time to form For convenience in writing, let U = y | = u/Ou. (6) 64. Complete Expansion of Exact Solution. — If. now, the substitution e ±w = e ±w+#o m € ±<*i ( cos ft ± j sin ^) ( 7 ) be made in equation (3), we obtain E = \ (E + Uh) (? l (cos 01 + j sin 01), + H^o - tf/o) c- a/ (cos 0Z - ; sin #) / = ^(l o -\-jjE^ l (co8 0l+jsm0l) + | (l - jj E)j e-^fcos 01 - j 'sin 0Z) . (8) Equation (8) is particularly well adapted to the physical inter- pretation of the mathematical expressions, although, as in the solution of the problem for the transmission of direct currents over leaky lines, a more convenient form for computing purposes may be used. 65. Interpretation of the Various Terms in the Expanded Solution. — In equation (8) consider first the quantity cos 01 ± j sin 01 = «*** INTERPRETATION OF VARIOUS TERMS IN SOLUTION 99 The absolute value of this expression, as the square root of the sum of the squares of the two components, is alway unity, so it is evident that the only function such a factor can perform is to rotate, or change the phase position of, a vector representing an alternating quantity. In fact this quantity is called a "rotating operator," because it rotates any vector which is affected by it as a multiplier, through an angle pi. cos pi + j sin pi = e+#" rotates a vector in a counter-clockwise direction, and cos pi — j sin pi = e~#' rotates a vector in a clockwise direction — in both cases by an angle numerically equal to pi radians — while the length of the vector remains unchanged. Returning to equation (8) it is seen that the line voltage (the same remarks apply to the line current) consists of the vector sum of two apparently distinct components. For increasing values of I, that is, going from load towards generator, the vector representing the first component, E' - | (#„ + UI ) e l (cos pl + j sin pi), (9) rotates in a counter-clockwise direction by an amount proportional to Z, and at the same time increases in length on account of the factor e"*. Counter-clockwise rotation of a vector signifies an advance in phase position of the harmonically varying quantity represented conventionally by this vector. Since the variation of e' with time is simple harmonic at a fixed point in the line, and for different points along the fine the phase position of e' advances proportionally to the distance, it appears that this quantity E' is merely the vector representation of a voltage wave moving in a negative direction along the line (generator towards load) and decreasing in magnitude as it moves along. Since E' at points near the source of power is advanced in phase position with respect to E' at points more remote, it is obvious that the direction of the wave motion of this component is opposite to our arbitrarily selected positive direction along the line. For increase in Z, this component increases, but speaking of the wave represented by E', it may be said to decrease in magnitude in the direction of its propagation. 100 ALTERNATING CURRENT PHENOMENA As to the remaining component, E" - | (E - l//o) e-" 1 (cos /M - j sin 01), (10) it is seen that the vector which this represents rotates in a clock- wise direction as I increases, and at the same time decreases in magnitude according to the multiplier e - "*. E" is thus the vector representation of a wave moving along the line from the load towards the generator — decreasing in value as it goes. According to the above physical interpretation of the meaning of the two terms in the expression for the line voltage, the e.m.f. at any point is made up of the sum of two separate waves moving in opposite directions; a main wave, £", moving from generator towards load, and a reflected wave, E" , moving from the load towards the generator. Both of these waves decrease in magni- tude at the same proportionate rate, in the direction of their propagation. This decrease in amplitude of each of the separate waves is due to the loss of energy in the resistance and leakage of the line, that is, the vhr and e 2 g losses. For a line with no resistance or leakage the waves do not change at all in magnitude, though the effective line voltage or current of course varies along the line on account of the different phase positions at which the main and reflected waves combine into the resultant. As stated before, the current equation may be interpreted in exactly the same manner as has been done with the expression for v6ltage. The two components of the resultant current are r = l( h + b Eo ) ** 1 (cos fil + j sin fil)t (10a) which represents the main wave, or wave traveling from generator towards receiver, and 1" = l(lo-jj Eo) r* (cos 01 - j sin 01), (10b) which represents the reflected wave, traveling in the opposite direction from the main wave. 66. Example of Calculation of Fundamental Constants in the Equations. — Before going further with a discussion of the equations, a numerical example will be considered in order to illustrate the method of determining the numerical values of the quantities which enter into the equations and to form a basis for the construction of the vector diagrams representing the phenomena. EXAMPLE OF CALCULATION 101 TABLE XII Calculation of Auxiliary Constants of Transmission Circuit Fundamental line constants, per mile to neutral: r = 0.275 = 0.15X10-°, L = 0.00204, C = . 0146 X 10-*, / = frequency = 60 cycles , Log L = 7 . 309630-10, Log C - 2 . 164353-10. X = 2ir/L log/ log 2 7r/ log a; logr 1.778151 2.576331 9.885961-10 9.439333-10 X tan 6 Z = - r log tan 6 Z log sin Z 0.446628 9.973872-10 X log z 0* 9.912089-10 70° 19' 26.42" Z = r +jx = Z/Jz sin Z b = 2tt/C log 6 log? 4.740684-10 3.176091-10 6 tan 6 y = - 9 log tan 6 y 1.564593 log sin U 9.999839-10 b y = — — sin y logy 0y 4.740845-10 88° 26' 20.16" Y = g+jb = ylh v 2 = zy 2 log v 4.652934-10 V 2 = ZY = zy/dz + By 2 Ov — 6z ~T uy 2 0„ 0„ log i> 158° 45' 46.58" 79° 22' 53.29" 7.326467-10 V = a+j0 = v/e v a = v cos 6 V = v sin 0„ log cos 6 V log sin 0„ log a log/3 . 360 log o- 9.265453-10 9.992498-10 6.591920-10 7.318965-10 1.758123 «°_ 360 * log/3° 0° 9.077088-10 Q. 000390769 0.119423 «*«-* y 2 log u 5.171244 tf-Vf- ■■ u/0 u 2 0u = 0« — 0y 2 0„ log M U l0g ^ -(18° 06' 53.74") 2.585622 - (9° 03' 26.87") 7.414378-10 b-fz- ■;*=* U 385.143/- (9° 03' 26.87") 1 E7 0.00259644/9° 03' ! 26.87" 102 ALTERNATING CURRENT PHENOMENA The numerical calculation of the line constants is given in detail, and, as is most convenient, is carried out by means of logarithms. Even when carried out with the accuracy afforded by six-place logarithm tables the amount of work required is very small, no work being required other than that indicated in the table. In general it would not be advisable to make such a computation with six-place tables, since the accuracy of the computation is very much greater than that of the fundamental data. Five places are usually sufficient. 67. Vector Diagrams Representing the Exponential Form of Equations. — Suppose that at the load the e.m.f. between one wire and neutral is 50,000 volts, and that the load current is 25.0 amperes. Suppose the power-factor of the load to be such as to cause the current to lag 25.0 degrees behind the voltage in time- phase position. Using the numerical values of the constants as determined from the calculations in Table XII, with equation ■Fig. 22. — The Vectors Representing the Main, Reflected, and Resultant Voltage Waves at the Load End. (8) the vector diagrams representing the separate components, as well as the resultant, of the voltage and current along the line will be constructed. If the load voltage be selected as the reference vector (that is, the arbitrary origin of time so selected as to make E a cosine variation), we have E = 50,000/0^ volts, 7 = 25.0 / — 25° amperes, and thus Uh = 9628.58 /- (34° 03' 26".87) vo it Sj jj E = 129.822 /9° 03' 26".87 amperes. Fig. 22 shows the vectors, E , h, Uh, and the two vectors representing the component waves at the receiving end, 2 Eo' = E + Uh and 2 E " = E - Uh. VECTOR DIAGRAMS 103 Fig. 23 shows the analogous current vectors, with E again shown as the reference, E , I , •= Eo, and 2/,' = /„ + ^o and 2/o" = h-jjE*. If now a numerical value be assigned to I, say 100 (miles), the vector 2 E ' will be rotated in a counter-clockwise direction Fig. 23. — The Vectors Representing the Main, Reflected, and Resultant Current Waves at the Load End. through an angle fil, or in this case 11.9423 degrees, and at the same time increased in length by the factor e l = e O.039O769 = 1.03985. 2 E a " rotates by an equal amount in a clockwise direction, and decreases in length by the factor r* 1 = 0.96168. One-half the vector sum of these two vectors is the vector repre- senting the actual line voltage at the distance of one hundred miles from the load. This resultant is very easily obtained by drawing a vector from the origin to the mid-point of the line joining the extremities of the component vectors. In Fig. 24, E is the load voltage, to scale; 2 E<f and 2 E " are twice the load-end values of the main and reflected waves, respec- tively. The curve, Locus of 2 E', indicates the path followed by the end of the vector in question for continuously increasing values of I. For each 400 mile point in the line, the vector is drawn in the diagram. The last vector so drawn is for a line 2000 miles in length. For I = 2000 the rotation of 2 E' is 238.846 degrees, and the factor (? l is 2.1848. The curve, Locus of 2 E", indicates the path followed by the end of the vector 2 E" for continuously in- creasing line lengths. The spiral is in this case a decreasing one, and for I = 2000, the factor r" 1 = 1Q . Q = 0.45770. 2.1848 104 ALTERNATING CURRENT PHENOMENA The resultant of the two waves, E' and E" is given by the vector E, the end of which moves along the curve, Locus of E. The rotation of the vector E is not proportional to the line length as is the case with the separate components, since it is Locnsof2E' Fig. 24. — Diagram of Voltage Vectors for Increasing Line Length. Subscripts Refer to Line Length. made up of the vector sum of two vectors rotating in opposite directions and changing in length as they rotate. Fig. 25 contains the current vectors, all of which are treated in the way just described for the voltage vectors. For the load selected as illustration, the e.m.f. UI , which is added to and subtracted from the e.m.f. E in order to form the initial vectors VECTOR DIAGRAMS 105 for the separate waves, is considerably smaller than E , and there- fore the initial vectors E ' and E Q " do not differ very much from E . Thus, for distances up to approximately 100 miles from the load end, the resultant voltage vector does not change very much. Loons of 21 ! Fig. 25. — Diagram of Current Vectors for Increasing Line Length. Subscripts Refer to Line Length. In the case of the vector diagram of currents, the component Yj Eq is several times the load current, so that the current at the receiving end, 7o, is made up of the sum of two vectors nearly 106 ALTERNATING CURRENT PHENOMENA equal in length and displaced from each other by approximately 180 degrees. The rotation in opposite directions of these two component vectors produces a resultant vector / which varies very rapidly for small increase in line length, as may be noticed in Fig. 25 from the manner in which the curve, Locus of /, starts from its initial point at 7 . 2000 Fig. 26. — Voltage and Current Vectors. Numbers Indicate Line Length. The two loci of the ends of the resultant voltage and current vectors are reproduced in Fig. 26, the vectors for the end of each 400 mile section of line being drawn. For any point in the line, the power-factor is given by the cosine of the angle between the vectors representing E and I for this length. For example, in Fig. 26, where the numbering indicates the length of line to which each vector belongs, for the end of the 800 mile section the current OPERATING CHARACTERISTIC FROM VECTOR DIAGRAMS 107 is ahead of the voltage in time-phase position by an angle of approximately 10.5 degrees, while at a point 1200 miles distant from the load the current is lagging by 22.8 degrees. Again, at 2000 miles, the current leads by 25.8 degrees. For increasing line length this power-factor angle oscillates about an angle equal to that of the quantity U = y y, as will be shown later. Since the load as well as the line losses must be supplied by an average flow of power along the line in a negative direction (generator towards load) it is evident that this phase angle can never be greater than 90 degrees, for then the average power transmitted (EI cos <£) would reverse in sign, and thereby indicate a flow of energy in the opposite direction. In fact, for any line with losses (r or g present) the power-factor angle can never equal 90 degrees, for then there would be no average power passing the point in question to supply the line losses in the portion of the line more distant from the generator. 68. Operating Characteristics from Vector Diagrams. — From the original drawings for the preceding figures, in which the vectors were inserted for every 200 mile section, the following values were obtained: TABLE XIII Distribution of Electrical Quantities Along a Long Trans- mission Line Length, miles E M ., volts I M ,, amp. Power-factor angle, degrees K.W. per phase Effic, per cent 60,000 25.0 25.0 lag 1130 100.0 200 48,900 49.5 58.5 lead 1245 90.9 400 40,500 94.5 65.0 lead 1612 70.1 600 28,900 128.0 49.8 lead 2385 47.3 800 24,300 143.2 10.5 lead 3410 33.2 1000 34,700 139.5 19.5 lag 4550 24.8 1200 49,200 122.8 22.8 lag 5550 20.4 1400 60,600 107.0 8.5 lag 6400 17.7 1600 65,700 112.5 13.2 lead 7250. 15.6 1800 65,000 140.0 26.2 lead 8150 13.9 2000 61,700 173.5 25.8 lead 9620 11.7 (The above values were determined by measuring the vector diagram, so may be slightly in- accurate.) The above numerical results are plotted in Fig. 27, with line length as independent variable. 108 ALTERNATING CURRENT PHENOMENA 160 no uo U> I .SI oo •to 20 K.y : J Dea. 70 60 \i 50 (SO 10 30 Ox i has. \a igli 20 E/ 10 -10 I'M -SO .80 20 -40 ■^p< ►we -60 10 K.W. x io- 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Line Length, Thousands of Miles 2.0 Fig. 27. — Voltage, Current, Power, and Phase Angle in a Long Line. 69. Limiting Forms of Vector Diagrams for Infinite Length Line. — In this figure the curves representing E and I approach the simple exponential curve in form when the line length becomes infinite. This follows from the disappearance of the term repre- senting the reflected wave, in equation (8), because of the de- creasing exponential. The equations of the limit-curves are thus E = Eo'r l , and I = /oV, in which scalar values are used. The wavy appearance of the curves in Fig. 27 for the line under consideration is due to the combination of the main wave with the reflected wave, which, for the comparatively short length of line, has not disappeared. The power curve, on the other hand, does not approach a simple exponential curve as a limit, as will be shown under the discussion of power relations and line losses. 70. Instantaneous Values from Vector Diagrams. — If it is desired to know the instantaneous distribution of current and e.m.f. along the line it is, of course, only necessary to draw the circles on the vectors in the diagram as diameters, and then to INSTANTANEOUS VALUES FROM VECTOR DIAGRAMS 109 measure the intercept on the time radius drawn through the origin at an angle with the reference vector equal to the product of the angular velocity of alternation (co = 2rf) and the elapsed time between the zero instant and the instant for which the distribution is desired. If the diagram represents effective values, the inter- cepts must be multiplied by the square root of two in order to secure instantaneous values. Kilo- volts 80 e. 60 /> 40 20 e" Lin c r. me li 11 D idtf '(IS ( )fJI ilea 1 - ( 1 **d 1 1 ) i > 1 4 1 .1 S 1 1 e£. -20 e^, -40 V e, -60 -80 Fig. 28. — Instantaneous Distribution of Voltage 0.075 Second After Positive Maximum of E . By selecting E as the reference vector in this numerical illus- tration, we have, as functions of time, e = a/2 50,000 cos 2 rft, referred to the instant at which e passes through its positive maximum, and U = ^2 25.0 cos (2tt/Y - 25°). Let the curves be plotted showing the instantaneous distribution of current and voltage 0.075 second after the origin of time. The time radius will have turned through 0.075 X 60 = 4.50 complete revolutions, or will be displaced from the reference line by an angle 110 ALTERNATING CURRENT PHENOMENA Amp. /' TOO / 120 80 4" 40 I.i It' I «M -tli III ndiBda of Mila 1 1 1 S 1 j ' 1 1 1 2 1 I 16 18 20 J i" -40 -80 \. -120 Fig. 29. — Instantaneous Distribution of Current 0.075 Second After Positive Maximum of E . Watts Vul{ at-'t'^ '"' V X vr* J f „-~ — 4j« rrer t S \ 14 y N v v \ > 12 / / \ V 10 / r Poa er \ 8 / 7 6 / >■'/ 1 4 • < 2 s / 1 y /\. lie .('11 .Mil. Hu uln ■,1s ( <t 1 Dm 1 J^ y 1 / \ I 1 1 1 1 4 1 ; i - •> °-2 **" / -4 / / -6 / / 4 / / • / t y Fig. 30. — Instantaneous Distribution of Power 0.075 Second After Positive Maximum of E a . Voltage and Current Curves Reproduced from Figs. 28 and 29. LOCI OF THE ENDS OF THE COMPONENT VECTORS 111 of 180 degrees. The square root of two times the intercepts of the various circles on this radius gives the values which are plotted in Figs. 28, 29, and 30. 71. Instantaneous Values of Power at Different Points. — In Fig. 30, the instantaneous values of power are obtained from the products of the corresponding instantaneous values of current and e.m.f. It is interesting to note that at this particular instant, for the section of line between I = 600 and I = 800 miles, the power is negative — that is, the flow of energy is from load towards generator — while on either side of this section the energy flow is from generator towards load. At any point in the line at which the power-factor is not unity, these negative values of instantane- ous power may be observed at some instant. From Fig. 27 it is seen that at the distances of 40, 870, and 1480 miles from the load the angle of lag is zero (power-factor unity), and thus the in- stantaneous power at these points cannot be negative at any time. These large variations in the distribution of instantaneous power along the line are of course occasioned by the continual and periodic redistribution of the energy stored in the line self-inductance and electrostatic capacity. To show the variation in power distribution along the line from instant to instant, Fig. 31 contains curves giving the instantaneous values of power for six successive intervals of time, beginning with the instant at which E passes through its positive maximum (t = 0) and covering one-half of a cycle. No. 0, t = 0. .No. 1, t = 1/720 second, 1/12 cycle. No. 2, t = 2/720 second, 2/12 cycle. No. 3, t = 3/720 second, 3/12 cycle. No. 4, t = 4/720 second, 4/12 cycle. No. 5, t = 5/720 second, 5/12 cycle. Though this set of curves covers only a half-cycle of voltage or current, it is sufficient to cover a whole cycle of the power wave, since this latter is of double the frequency of its two components, voltage and current. 72. Equations for the Loci of the Ends of the Component Vectors. — The curves traced out by the ends of the vectors which represent the component waves are logarithmic, or equi- angular, spirals. Take, for example, the locus of the end of the 112 ALTERNATING CURRENT PHENOMENA vector representing the main wave. Its initial length is eo', and for increasing distance from the load end of the line, its length is represented by the equation e' = eoV, (11) fa 20 Y 18 X 16 11 0/ 12 ^\ 10 V 8 I JL 6 •V 4 2 -2 -4 Dis tuiu o f r Dill Lou din Hu mlr eils rfl files . -6 ■> 4 8 10 12 u 10 18 20 Fig. 31. — Instantaneous Distribution of Power 0, 1, 2, 3, 4, and 5 Twelfths of a Cycle after the Positive Maximum of #0. Numbers on Curves Indicate Twelfths of a Cycle. and the angle the vector makes with its initial position is e = 0Z. Eliminating the parameter I by I = e' = e»'f\ (12) (13) which is the equation of the exponential curve in polar coordinates. Let <f> = the angle between the tangent to the curve and the radius vector. Then from the geometry of the figure, ^ = ecot*. (14) CONSTRUCTION OF EQUIANGULAR SPIRALS 113 By differentiating (13), de' , a ^e a . ,,_,. *-*V =r~ (15) so that by comparing equations (14) and (15), there results cot« = ~ (16) From this it is seen that cot 0, and therefore <j>, is a constant. It is from this property that the curve derives the appropriate name of "Equiangular Spiral." In the expression V - VZY = v/dy and, as previously shown, a = v cos 0„, j8 = v sin 0„, so that - = cot 6 V . Thus, from (16), the angle <f> between the tangent to the curve and the radius vector is the same as the angle 6 V , which, in turn, is equal to the mean of the impedance and admittance angles. = 0„ = H0. + 0v). (17) For the decreasing spiral, the angle is the same as above — the rotation of the radius vector in the opposite direction causing a diminution instead of an increase in the length of the radius. 73. Mechanical Construction of Equiangular Spirals. — The equiangular property of these curves suggests the possibility of constructing them by some purely mechanical means. This may be done by the use of the apparatus shown in Fig. 32, in which a small sharp-edged tracing wheel is mounted in a frame so that the plane of the wheel may be set at any desired angle <f> with the axis of the instrument. Along the axis of the instrument is a narrow slit, which slit slides over a pin placed at the center of the vector diagram, at 0. A scale of degrees and a vernier may be placed on the rotatable mounting of the tracing wheel so that the angle may be set off accurately. The zero of the scale may be located by 114 ALTERNATING CURRENT PHENOMENA ascertaining by trial the position at which the wheel is to be set so as to trace out a circle. By pressing the wheel firmly against the paper and rotating the entire apparatus about the point 0, an equiangular spiral will be traced out as shown — the angle <j> at which the instrument is set being the same as the angle <f> in equations (16) and (17). As shown, rotation in a counter-clockwise #-» B \+(*«+M Fig. 32. — Polar Exponential Curve Tracer. direction gives the increasing spiral, and rotation in the opposite direction, the decreasing spiral. Using such a device, it is only necessary to draw in the diagram the two initial vectors, 2 E ' and 2 E ' ', and with the angle <f> set off on the vernier and scale, to place the tracing wheel upon the end of each vector and to draw in the spirals. This done, the vectors, 2 E' and 2 E", corresponding to any particular line length are obtained by drawing the lines which make angles /3Z with the initial vectors, from the center of the diagram to the curves. The resultant vector is obtained as before, by drawing the line from the origin to the mid-point of the line joining the extremities of the two component vectors, 2 E' and 2 E" . By the use of such a device, comparatively accurate solutions for lines of different length, but with constant receiver load, may be obtained very rapidly after having at first computed the numerical values of the auxiliary constants, a, /3, and U, as per Table XII. 74. Introduction of Hyperbolic Functions of Complex Vari- ables. — In the preceding section the manner of the variation of the line phenomena with change in line length has been dis- cussed. The vector diagrams, for any distance from the receiving end of the line have been shown to be very easily obtained by the use of a mechanical device for tracing the exponential curves. CALCULATION OF CONSTANTS IN EQUATIONS 115 Such procedures are particularly applicable to cases of constant receiver load and variable line length. In dealing with a line of given length over which a variable load is to be supplied, a more convenient form of expression than the exponential in equation (8), as exemplified in the vector diagrams just given, may be obtained by using hyperbolic functions. By combining terms in equation (3), the complete solution may be put into the form (18) E = E \ (^ ZYl + e^™) + Jo\/f \ (* VZYI ~ e~ VzYl ), I = 7o \ (e V ™ + e-^0 + %>SJ\ \ (e V ™ - e" V ™), ( which, by introducing V=VZY, and U = \Jy, and the hyperbolic functions for the combinations of exponentials, become E = E cosh Vl + hU sinh VI, I = h cosh Vl + E jj sinh VI (19) Naturally, equation (19) is identical in form with equation (14) of Chapter II for the direct current system. V = VZY is of course complex, so that the hyperbolic functions of a complex quantity are required. V = a+jfi. cosh VI — cosh al cos 01 + j sinh oil sin f$l, | sinh VI = sinh al cos fil + j cosh al sin /3Z. J Equation (19) for the alternating current problem may be carried through all the developments as are given in Chapter II for the direct current system, keeping in mind, of course, the fact that the complex quantity notation must be retained. 75. Calculation of Constants in Equations for a Line of Given Length. — The utmost convenience and dispatch are to be found in the use of equations (19) for the determination of the operating characteristics of a transmission system, and particularly if a combination of analytical and graphical methods is employed. A numerical example will serve best to indicate the method. 116 ALTERNATING CURRENT PHENOMENA Let it be required to plot curves showing the operating char- acteristics of a transmission line whose constants are those given in Table XII and whose length is 400 miles, for the condition of constant voltage at the load end. The first step in the work is to obtain the numerical values of the three coefficients cosh VI, U sinh VI, and jj sinh VI, as required in equation (19). From the values of a and /3 given in Table XII and using tables of the logarithms of hyperbolic functions, the computations may be conveniently carried out as shown below in Table XIV. TABLE XIV. Computation of Constants in Hyperbolic Expressions for Transmission Line Phenomena Line length = 400 miles From Table XII, a a al f* log cosh al log sinh al log sin pi log cos pi log A c = log cosh al cos (31 log B c = log sinh al sin pi log tan B c log cos B c log cosh VI log A, = log sinh al cos pi log B, = log cosh al sin pi log tan B t log sin B, log sinh VI From Table XII, log U log U sinh VI log jj sinh VI cosh VI U sinh VI Results of Computation jj sinh VI 0.000390769 0.119423 degrees 0.156308 47.769 degrees 0.00528 9.19575-10 9.86949-10 9.82745-10 9 . 83273-10 cosh VI = A c +jB e 9.06524-10 9.23251-10 9.99376-10 (9 . 83897-10) 79.693 degrees 9 . 02320-10 si nh VI = A . + jB. 9.87477-10 0.85157 9.99574-10 (9 . 87903-10) /81 .989 degrees (2 . 58562) /- 9.057 degrees (2 . 46465) 772.932 degrees (7 . 29341-10) 791.046 degrees . 69020 /9.693 degrees 291.51 /72.932 degree s ohms. 0.0019652 /91.046 degrees mhos. Thus, numerically, E = E [0.69020/9^693] + 7 [291.51/72^932] I = I [0.69020 /9°.693 ] + E [0.0019652 /91 °.046] GRAPHICAL TREATMENT FOR LOAD-END CONDITIONS 117 76. Graphical Treatment for Various Load-end Conditions. — From this point on, the remainder of the solution may be obtained graphically by constructing the vector diagrams representing equation (19) on the basis of the numerical values furnished by Table XIV. Since the load voltage is supposed to be kept constant, the vector diagram is constructed on E as reference E for P.F. Angle »0 and I -= 100 Amp, I U «inh VI ^ Voltage Scale ! — I 1 1 1 1 1 1 1 1 1 10000 20000 Fig- 33. — Voltage Diagram for Various Load Currents. Based on Equation (19). vector. Let E — 50,000 volts, effective. Fig. 33 is the voltage diagram, representing the equation E = E cosh VI + IqU sinh VI. The resultant voltage at the generator is made up of two parts — one part proportional to the load voltage and the other to the load current. The vector representing E cosh VI = 34,510 /9.693 deg. is drawn at an angle of 9.693 degrees from E and with a length to the same scale as E representing 34,510 vols. This vector, which is one of the two which go to make up the resultant vector for E, remains fixed regardless of the load current 7 . In the diagram, A designates the end of this vector. To complete the diagram and make it applicable to any and all 118 ALTERNATING CURRENT PHENOMENA values of load current at any power-factor, select a base value of current of, say, 100 amperes at unity power-factor. Then h = 10 0/0 deg., and IoU sinh VI = 29,151 /72.932 deg. volts. The resultant generator voltage at E is obtained by adding the two vectors in the ordinary manner, giving, for this particular load, E = 54,30 0/38.2 deg. by measurement of the diagram. This gives the solution for only one value of load current, but it is readily seen that for any other value of load current at this power-factor, the solution is obtained by taking the proportional value of the vector I U sinh VI, and, of course, of AE. For example, the line AE may be divided into five equal parts, giving thereby the values of generator voltage for I = 0, 20, 40, 60, 80, and 100 amperes. An extension of the vector beyond the length of AE gives solutions for load currents greater than 100 amperes. The length of the vector I U sinh VI depends only upon the absolute value of I , while the angle it makes with the reference line OE depends only upon the power-factor of the load, and is, in fact, equal to the sum of the power-factor angle and the angle of the factor U sinh VI. Thus, for any power-factor of load, it is only necessary to turn the vector I U sinh VI, as drawn in the diagram for unity power-factor, through an angle equal to the load power-factor angle. Rotation of this vector also rotates the line AE through an equal angle. The dotted lines in the figure indicate the vectors representing the solution for a load current of 100 amperes lagging 25 deg. behind the load voltage. Arcs of circles drawn through the points subdividing the line AE provide solu- tions for the intermediate values of current at any power-factor. To obtain the value of the generator current we again take the sum of two components — one proportional to Eq, and therefore constant in this particular case, and the other proportional to 7 . Fig. 34 is the diagram of current vectors with E again used as reference. The quantity #077 sinh VI required by equation (19) is equal to (50,000 /0 deg. ) ( 0.0019652 /91. 046 deg. ) = 98.26 /91.046 deg. amperes. GRAPHICAL TREATMENT FOR LOAD-END CONDITIONS 119 This current is represented by the vector OA. To it is to be added the remaining component I cosh VI which, in this case, depends upon the load current and the power-factor. Select a base value I for P.F.Angle«0 and I ~100 Amp. B^sinhVlefZZ Fig. 34. — Current Diagram for Various Load Currents. Based on Equation (19). of current of one hundred amperes at unity power-factor as was done in constructing the voltage diagram. Then 7o cosh VI = (100/Odeg.) (0.69020 /9.693 deg. ) = 69.0 2/9.693 deg. amperes. The vector representing this quantity is drawn to scale in the diagram, making an angle of 9.693 degrees with OEq. The result- 120 ALTERNATING CURRENT PHENOMENA ant generator current for 100 amperes load current at unity power-factor is given then by the vector sum at 01, and is I = 128.3 /58.8 degrees amperes. By a method identical with that followed for the voltage diagram, the generator current for any value of load current at unity power- factor is obtained by taking the proportionate value of the vector AI. If the load power-factor changes, the vector A I is rotated through an angle equal to the power-factor angle, so that solutions for all values of load current between and 100 amperes are deter- mined by the intersections of the circular arcs corresponding to the particular numerical value of current and the line drawn from A making an angle equal to the power-factor angle with A I. The dotted lines are the vectors for a load current of 100 amperes, lagging 25 degrees behind the load voltage. 77. Power-Factor from Vector Diagrams. — The power-factor angle at the generator is equal to the difference between the angles which E and I make with E . Thus, where E = 54,30 0/38.2 deg. and I = 128.3 /58.8 deg. the difference in the angles is 20.6 degrees. The angle of I is the greater, and thus the current is ahead of the voltage in time-phase position. The power-factor at the generator is cos 20°.6 = 0.936. The volt-amperes at the generator = ei, and the power in watts = ei cos 6. 78. Operating Characteristics of Line with Various Loads. — In Table XV are collected a number of numerical values, taken from the vector diagrams by measurement, to illustrate the characteristics of the line under consideration. The values are all based upon a constant load voltage of 50,000 volts to neutral. Three power-factors at the load are considered, 25 deg. lagging, deg. lag, and 25 deg. leading. Since the numerical values are obtained graphically, the last significant figure may be slightly in error. The curves in Figs. 35, 36, 37, 38, 39, and 40 are plotted from the values tabulated in Table XV. As would be expected, the generator voltage increases most rapidly (to maintain E constant) in the case of the lagging load, and least rapidly for the leading load. In fact, the generator voltage nearly doubles in the former case, while in the latter case the increase is only 33 per cent for a range of 7 from to 100 amperes. OPERATING CHARACTERISTICS OF LINE WITH LOADS 121 K.V. 60 No. 1, ~5 i Lagging Loin No. 2, Of {' I " . 58 No. f, ts Leading IjOUc] 56 54 1 52 50 48 46 \s 44 42 40 J*> 38 36 34 32 ( 30 10 20 80 90 100 30 40 50 60 70 Load Current, I , in Amperes. Fig. 35. — Generator Voltage Required to Maintain E Constant at 50,000 Volts. 150 „ ^< .mp, 110 Ncl. 1, 25" I Nd.2, |o° Bffg uk' I -.oail No .3,2 l»l eadi ngl oad 8 >X 130 120 2 110 100 _J_ 00 80 Pig. 36. 20 80 ;>o 100 80 40 50 60 70 Load. Current, Jo » in-Ampcres. Generator Current for Different Load Currents with E Constant at 50,000 Volts. 122 ALTERNATING CURRENT PHENOMENA 80 Dq MM 70 60 ^N ^1 60 :1 s 40 No No 1,2. 2. fit «x iaii. No 3.2 ,'L. •ailii 1^' i- jad. 30 ^ 20 10 10 ■ SO M 100 30 40 50 60 70 .Load Current, 1 , in Amperes. Fig. 37. — Angle by Which Generator Current Leads Generator Voltage. Load Voltage Constant at 50,000 Volts. 10 30 BO BO 30 40 50 60 70 Load Current, I , in Amperes. Fig. 38. — Kilovolt-amperes at the Generator with Load Voltage Constant at 50,000 Volts. OPERATING CHARACTERISTICS OF LINE WITH LOADS 123 No. No. 1. 25 L Lagging "1 Loai No. \ 25 Lea ling Leai . 3 /\ I oat Cu rren M , in An per 3S. 10 20 1 T f 50 T f 80 T lOo K.W. 7000 6000 5000 4000 3000 2000 1000 Fig. 39. — Power at Generator for E Constant at 50,000 Volts. 80 h-* i 2 70 T~" 60 50 40 No. No. 1,26" '-. Lagging Load " r- No. i, 25 Lea ling Load L 20 10 - 10 30 40 50 60 70 Load Current, I ft , In Amperes. itO 100 Fig. 40. — Efficiency of Transmission for E Constant at 50,000 Volts. 124 ALTERNATING CURRENT PHENOMENA > X w i ! E . & § 5 ii ■3 © x II H S 00 3 o o e 5 s - ««< CM ■»* CO CO CO oooo OcpCNOOgitN O oo £ co in —c CM CO ■* O 3< t- CM < <N CM CM CO (NO 288888 >o <N CO 35 O0_tN ^O^O-'clfd' coco-3"* ic © t^t^. CM CO -h i-i co in t- oo o od>d>d>S<d> eg co o CM CD <n m idO'tO CNOO •^•oiesiior^ « r-iCMcM CM 'tNirSxS r. C r r. co :g£288 HONC CM kQ — i CM CO 'O CO 35' co co ■*£ io o co CO CN-H ocot-cp 00^ oo COCMOOO 288SS8 in 35 cm cm r- co "0 co QOO^OCO CO «5 i-i CM CO t^- CO H^CONCCO oodddd g CO CO I— CM CO 'OOiONOO I^SoOO -h oo co oco CO COOOO O CD CM 00 JO CM O 35 CO ° 00 o CM-^OOOO od cbiouo dop OO'HNCO'}' 28885*; • O CM 00 CM CM ( 0.150: 0.416 0.627 0.780 0.883 0.947 co "0 CO -^ CM t- O 00 r-i in <- < oo oo oo oo co in co cm i-i 91.046 84.7 79.4 74.8 71.2 68.0 9.693 19.3 28.2 36.1 43.2 49.2 CNiCOXO CONSTANT GENERATOR VOLTAGE; RIGOROUS SOLUTION 125 In Fig. 36, showing the generator current, it is seen that at a load power-factor angle of 25 degrees lag, the generator current is practically constant throughout the 100 amperes range of load current. The total range of generator current is only from 94.2 amp. to 102.9 amp. — a maximum variation of 9.2 per cent. At this condition of load, the result of the power transmission is to effect a constant current to a constant voltage transformation. For in-phase and leading load currents, the generator current in- creases quite uniformly, though by no means as rapidly as the load current. As shown by Fig. 37, within the range of 25 deg. lag to 25 deg. lead, the generator power-factor depends only very slightly upon the load power-factor, but does depend upon the load current. The transmission efficiencies as shown in Fig. 40 reach a maxi- mum value in all three cases at a load current of from 70 to 80 amperes. As would be expected, the efficiency is greatest for the lagging load — the PR losses in the line being greatly reduced on account of the diminished resultant value of line current, the lagging component of the load current neutralizing, in part, the leading capacity current of the line. Of course, if the leakage losses in the line were considerably greater (increased conductance g), the increased E 2 g loss, on account of the greatly increased voltage required at the lagging load power-factor, would cause the efficiency in this case to be lower, and in extreme cases, lower, even, than the efficiency at unity and leading power-factors. 79. Constant Generator Voltage ; Rigorous Solution. — In the preceding section we have discussed a very convenient method for determining the performances of a transmission system when the - load voltage and load current are both known — the con- venient graphical processes being based upon the explicit equa- (19). If the problem be stated differently — say constant gen- erator voltage and known load currents — it would appear that no such convenient solution is possible. The knowns in such a case are load current and power-factor, and generator voltage. We are not, in fact, at liberty to specify both the load current and generator voltage throughout unlimited ranges, for we might select a value of load current greater than the generator voltage could supply over a short-circuit at the load, even. The second of equations (19), Chapter 2, when written 7o = Z coshV7 + C/sinh VV (21) 126 ALTERNATING CURRENT PHENOMENA to conform with the notation used in the alternating current problem, can be made to yield a solution, though not without involved expressions and tedious numerical operations. The procedure is as follows: the numerical values of E and J are known as scalars; e being the fixed generator voltage and i the value of the load current for which a solution is desired. The ratio of the two components of the effective load impedance Z is known, for it is this ratio which determines the load power-factor. The numerical value of '/... is unknown, and its determination con- stitutes the essential part of the problem, viz., the determination of the load voltage and the generator current. Let <fo = tan -1 — • Then the load current lags by the angle <£o behind the load voltage, and Zo = 2o cos <£o + jz<> sin <£o = Zo/^o, (22) where Zo is the unknown scalar value of Z . With Zo retained as an unknown and the angle <fo known, the product Z cosh VI = Zo (a + jb), in which a and b are known, and also (23) UsmhVl = c+jd is known, so that Zo cosh VI +U sinh VI = Zoa + c + j (z b + d). (24) E By (21), the scalar value of j- is equal to the scalar value of the ■to expressions in (24), so that (ff-(**+c)*+W>+dp, or zo 2 (a 2 + V) + zo (2ac + 2bd) + c 2 + & -(^\ = 0, which gives - (ac + bd) ± 20 ~ a 2 + 6 s y (ac + bd) 2 - (a 2 + 6 2 ) c 2 + d 2 - (£j (25) CONSTANT GENERATOR VOLTAGE 127 The positive value of z from (25) is to be retained. Failure to secure such a positive value indicates that a value of t'o has been used which is greater than the generator voltage e could supply. Having the numerical value of z , the load voltage is immediately- obtained by E = Zo (cos <£o + j sin <fr>) 7 o, where 7 may be used as reference vector. Then E and 7 are obtained by equations (19) in their proper complex quantity notation by using 7 and E as found. A good check on the accuracy of the numerical work is afforded by the fact that the recomputed scalar value of generator voltage must agree with the value assumed at the outset. Of course, it is not necessary to recompute the value of E unless it is desired to check the work or to secure values of generator power, in which case it is neces- sary to do so in order to secure the power-factor angle at the generator. 80. Constant Generator Voltage ; Solution by Use of Plotted Curves. — The procedure above outlined is not at all adapted to the computation of data for curve plotting, but it, or its equiva- lent, must be followed if results more accurate than could be read from curves are desired. If plotted curves are sufficiently accu- rate, then the solutions for constant generator voltage may be obtained with great ease by an indirect process, based upon the results obtained for constant load voltage in the preceding para- graphs. By equation (21) we see that 7 varies directly as E as long as the equivalent load impedance Z remains constant. E = ZoIq thus varies directly as E also. In Table XV, each solution as tabulated for a given value of 7 corresponds to some particular value of Zo, so that if we assume that the value of Z remains constant while the generator voltage changes from the tabulated value, obtained by the graphical solution described, to the constant value we wish to consider, the corresponding values of 7 and E for this latter case can be found by taking them in the same ratio to the original values as the ratio of the new E to the original E. The generator current will also change in the same ratio, while the values of power and volt-amperes will change by the square of this ratio. For example: in the solution for 7 = 80 amperes at a leading power-factor angle of 25 degrees and Eo = 50,000 volts, 128 ALTERNATING CURRENT PHENOMENA E = 42,250 /43.2 dcg. and / = 136.9 /71.2 deg. If E be changed to 50,000 volts, then by proportionality 7 = 4225? X 136,9 = 162 ^P 61 " 68 . 50 000 h = aoo-o x 80, ° =94 - 7 amperes. E = |^^ X 50,000 = 59,200 volts. Po =(^^) 2 X 3625 = 5080 K.W. p =(ti5) 2x5100 = 7140K - w - K.V.A. = (|f^) 2 X 5780 = 8090K.V.A. at gen. The efficiency, of course, does not change, since both load and generator powers change in the same ratio. It may be remarked here, as will be discussed later, that the efficiency of transmission of a given line depends only upon the equivalent load-end imped- ance Z , and not in any way upon the magnitude of voltage applied at the generator end of the line. Thus, in general, let K _ new generator voltage . . former generator voltage Then New currents or voltages = K (former currents or voltages), New powers or volt-amperes = K 2 (former powers or volt-) (27) amperes). In the transmission line under discussion let the generator volt- age be so adjusted that the load voltage is 50,000 volts at a load current of 80 amperes, lagging 25 deg. Its value from Table XV = E = 54,800 volts. Table XVI shows the method of ob- taining the data for performance curves under the condition of constant generator voltage from the data for constant load voltage in Table XV. The curves in Figs. 41 to 46 inclusive have been plotted with load current as abscissse, or independent variable, from the data in Table XVI. SOLUTION BY USE OF PLOTTED CURVES 129 Lmp. 180 Ji_ 170 160 150 2 140 130 120 ^1 110 No. l,2.-> "La ersir ?L 5aci 100 No. No. 2. 0« |" 3,25°LeadlE gr Load .90 80 70 60 Lot dC irre at I o,i nA mpc res 50 1 3 _J 3 3 * 3 5 61 | J 3 8 9 ° 1(0 Fig. 41. — Generator Current for Generator Voltage Constant at 54,800 Volts, Kilo- < rolts 80 75 ^8 70 65 60 \2 No. No. 1,25 .'. ;La K T.o;i( . 55 No. 3,25 Lea ling I. OP. I. \I 50 Los dC ilT( nt, Co, 1 i A ape res. 45 T I 30 o T " ik 8. " 100 Fig. 42. — Load Voltage with Generator Voltage Constant at 54,800 Volts. 130 ALTERNATING CURRENT PHENOMENA T"~ :.' No. No. i". uig ring Load So. {U u ling 1,oiW . Lo id c urr ?nt I... n^ mp< 2rcs 1 1 " T f 60 T i EI T wo K.V.A. 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 Fig. 43. — Kilovolt-amperes at Generator for Constant Generator Voltage of 54,800 Volts. 10 20 30 40 50 60 70 80 90 100 Load Current, I , in Amperes. Fig. 44. — Power at Generator for Constant Generator Voltage of 54,800 Volts. SOLUTION BY USE OF PLOTTED CURVES 131 i.w 7000 No No. 1, 25 " LafjBiut' Load 2. Q° | S | P 6000 No. 3, 2E »& adii gL( xul 3^ 50O» S-- 1000 1 3000 3000 1000 Lpad,< Sir •out I„ iD / Lm sere 9 T 20 1 T 40 1 50 60 T 80 T 100 Fig. 45. — Power at Load End with Generator Voltage Constant at 54,800 Volts. Jl 1 „ ~~~^ 7R ,70 i— ■- 60 50 40 30 No. 1, 25 No. 2, Lag [tag Loai . No. !, M 'Lea 1 1 rifjr Loft . 20 10 I 1 1 1 ) 1 ) 4 ) a ) a i 7< I 81 ) n i K Load Current, I , in Amperes. Fig. 46. — Transmission Efficiency with Generator Voltage Constant at 54,800 Volts. 132 ALTERNATING CURRENT PHENOMENA 1 O > I -r 9 s ■n X 3 < i 3 X _ JS o .fi > H 8 £ II ■Sbeq « . "3 3 tX ■ o .a ^ 8 a O O _ f* a CJ 3 °" a. a ■9 > * « ;£ &a 3 o "9|5q II -^(NrJ(«0«0 Soo c-i ci >o »-iC* co co co COQQOQO 00 »ft lift C5 Q ^h t~05O5O5 00 CO "J N °2 *2 92 "- 1 :s: 005«MN<0 00 C5 CO O Q ^h 28SS LOIN CO i c. r ■/■ r r OOQOOO co 35 -h t- o <N CO-* ■* "3 ift i-H t» -^ c5 -* 00 00 t~ t>. t» t» HO— <i NNNNO 00500C5 (ONHtDHft ■ft ■* ■«* co co oi t^-QOO^O OOOMhO-h !38 ii»eo i-ioocooeo Ooot^o^o 188° 'OQ-JONO /: _. — ~. C". Sa ,s 1-1 Ift 05 1^00 OOiftOCOOO b- >ft o oo co co '—' o 5 © SB i ift <M 55 iM (N I INFINITE SERIES TRANSMISSION LINE SOLUTIONS 133 81. Infinite Series Form for Transmission Line Solutions. — For short lines operating at low frequencies and having only moderate power losses, the solutions as given in equations (3) and (19) may be expressed by infinite series, the successive terms in which decrease rapidly. Equation (19) is perhaps more easily expanded than equation (3), so the former will be used. The series for the hyperbolic cosine and sine are coshz=l+g-+g4-^+. ' /y»o /j*5 T** sinhz = z + [ 3+g + g-+-. Introducing these series in (19) with x - VI = VzYi = Vziv~t, (28) where and we have immediately Z t Y t , Z?Y t * = #o(] = /o(l 2 Z t Y t 2A Z?Y? Z t = Zl, Y t = Yl, ••W5(- %YA (Z t Y t )i 6 • -Wit^-^ (29) 2 ' 24 which, by combining factors in the second portion of each, become Z t Y t E 2 ' + hZt(l + Z t Y t Z?Y? Zt z Y t 3 24 ' 720 ' Z t Y t , Z?Y? Zt z Y t 3 G 120 2 9 Y t ( z?y? zn? 24 "*" 720 + ' . ZtYt Z?Y? G 120 5040 ) Z t 3 Y t * 5040 + )] (30) Of course, Z t and Y t must be used in their complex form. The rapidity of convergence of these series is dependent upon the value of the argument Z t Y t) and thus upon the square of the product 134 ALTERNATING CURRENT PHENOMENA of the line length and frequency, and also, to a lesser extent, upon the product of total line conductance and total line resistance. The form of any type of solution is worthy of attention. The generator voltage, or current, is always expressed in terms of the load voltage and current by linear equations of the form E = AE + BIo,\ I = Ah + CE , J in which the constants A, B, and C are functions of the physical properties of the system and the frequency only, and not of the voltages or currents. In effect, all the various methods for trans- mission line solutions are only schemes for determining the values of these three constants. Having them, the solutions for various loads are obtained with as much ease in the case of a long line as in the case of a short one. In equation (31) the functions cosh VI, U sinh VI, and jj smn ^ are the values of A, B, and C, respectively. In equation (30), the series are again merely expressions for these constants. It may be remarked here that the current and voltage at any point in a complicated network of circuits containing but one source of energy can always be expressed as such linear functions of the voltage and current at any other point, as long as the physical properties of the different parts of the system are pure constants and independent of the voltage or current. By far the most advantageous procedure in the calculation of the performances of any such system is to calculate first the values of these constants and then to apply the resulting numerical expressions to the particular numerical values of E Q and 7 for which a solution is desired. Further, expression in this form always leads to the convenient graphical process of calculation as just described in application to the long transmission line. 82. Illustration of Convergence of Series. — To illustrate the manner of convergence of the above series expressions for A, B, and C, the following table shows their values, for the 400 mile line under consideration, when different numbers of terms are included in the parentheses. LINE ADMITTANCE AND IMPEDANCE LOCALIZED 135 TABLE XVII Convergence of Series Expression for the Constants A, B, and C in Solution for 400 Mile Transmission Line Number of terms used A B C 1 1.00000/0°. 000 326. 70/70°. 324 0.0022024/88°. 439 2 0.67732/11.093 290.53/73.124 0.0019586/91.239 3 0.69035/ 9.652 291.52/72.927 0.0019653/91.042 4 0.69020/ 9.694 291.51/72.931 0.0019652/91.046 Inf. 0.69020/ 9.693 291.51/72.932 0.0019652/91.046 An inspection of the tabulated values shows that, as far as engineering calculations based upon the more or less uncertain values of the line constants are concerned, the use of three terms in the series expression yields results, in this case, which are sufficiently close to the true values as given by the use of the hyperbolic functions (infinite number of terms). In the case of the B and C coefficients, two terms give very accurate results, but on account of the slower convergence of the hyperbolic cosine series, it is necessary to use at least three terms in the calculation of the coefficient A. In spite of the rapid convergence of these series expressions, the amount of labor involved in their use is much greater than for a rigorous solution based upon the hyperbolic functions directly, as in equation (19), provided tables of hyperbolic functions of real variables are available. Such being the case, there is no real need of setting up criteria for estimating, before the numerical computation is undertaken, the number of terms which must be used in the series to give a desired degree of approximation. The writer is well aware of the aversion of the general engineering public to the use of such hyperbolic function forms of expressions as equations (19), but, when such expressions are so easily dealt with numerically and at the same time furnish results concerning which there can be no doubt as to the accuracy, it appears that such prejudice is unwarranted. 83. Line Admittance and Impedance Localized. — From purely physical considerations, it appears that the effect of the distributed admittance of a transmission fine can be approximated by placing a lumped admittance across the fine at the mid-point 136 ALTERNATING CURRENT PHENOMENA of its length, equal in value to the total amount of the distributed admittance. Doing this, the equivalent circuit representing the transmission line is as in Fig. 47. In this circuit, E' = voltage at mid-point = E + 5 ZJ , I' - current through admittance = E'Y t = Y t E + £ Z t YJ . I = I + I' = I (l + lZ t Y t ) + E Q Y h ) W^W^r^VV^^ Fig. 47. — Approximate Representation of Distributed Line Admittance. In equation (32), based on the approximation that the admittance of the line is all concentrated at the middle, the constants in the general form of equation (31) have the values, A = {\ + \Z t Y t ), B = Z t (l + \Z t Y t ), C=Y t . (33) Comparing these values with the series, equation (30), we see that the expression for A consists of the first two terms in the series for the hyperbolic cosine. The expression for B is also the first two terms in the series for U sinh VI, except that the coeffi- cient of Z t Y t in the parentheses above is one-fourth, instead of one-sixth, as in the true series. The expression for C in (33) is equivalent to the series expression in (30) when the first term only (unity) is retained in the parentheses. The amount of compu- tation involved in using equations (33) is the same as in using the first two terms only of the rigorous series expressions (30), and the results are not as accurate, because the values of the coefficients in the B and C terms are not the same as in the series. Therefore there is no possible excuse for using the approximate expressions based upon a concentration of the total admittance at the middle LINE ADMITTANCE AND IMPEDANCE LOCALIZED 137 of the line, except, perhaps, in an academic way, to impress upon students the physical significance of leakage and capacity effects. Another approximation method is based upon a different con- centration of the line admittance. It assumes that the distributed line admittance may be replaced by locating two-thirds of the total admittance at the mid-point of the line, and one-sixth of the total at each end. This method is equivalent to assuming that the electrical quantities are distributed along the line in a way repre- sented by an arc of a parabola. (See Steinmetz' "Alternating W^W—r- W^fl^ Fig. 48. — Approximate Representation of Distributed Line Admittance. Current Phenomena," page 225.) Figure 48 shows the arrange- ment. Using the notation shown in the diagram, E 1 = E + IihZ t = Eo(l+f s Z t Y t ) + i ZJ . I" = | Y t E x = E Y t (f + ft Z t Y t ) + i Z t YJo. h = h + I" = E Y t (f + A Z t Y t ) + 7o (1 + | ZtYt). E^Ei + iZJ*. I = h + \Y t E. Zt 2 Y t 2 \ , r „ L . Z,Y t > E = £o(l+^-t 2 ZtYt 36 }+«* + ¥9 r-z.(x + ^ + ^)w,(x + |z.r. + ^)-| (34) Naturally, equations (34) are closer approximations to the series in equations (30) than are those developed on the assumption that the line admittance may all be concentrated at the middle. In the above equation, the expression for the coefficient A differs from that in the series by only Z?Y? 72 higher power terms, 138 ALTERNATING CURRENT PHENOMENA and is therefore quite accurate. The expression for the coefficient B does not contain the term in ZfYf, nor any above this order, but the coefficient of the Z t Y t term is the same as in the series. In the expression for the constant C the coefficients of both the Z t Y t and the Z t 2 Y t 2 terms are smaller than in the series. The labor involved in computing from equations (34) is the same as from the series expressions, when the same number of terms are included, so there is no reason for using the former, since they are not as accurate as the series when the latter are written to the same number of terms. To summarize, it is not possible to find any method of obtaining solutions for the type of alternating current transmission lines under consideration which surpasses, in ease of application, con- ciseness of expression, or in accuracy of results, the general and rigorous formulae expressed in terms of hyperbolic functions. Even the simplest approximation — single equivalent admittance at the middle of the line — does not materially lessen the work, and, as pointed out, is quite inaccurate — absolutely so when the line approaches a quarter-wave length. 84. Collection of Formulae for Use in Computing. Tabular Arrangement. — For the convenience of computers, the formulas and tabular form for the computation leading to the determination of the constants A, B, and C, in the general form of solution (31) through the use of the rigorous hyperbolic function expressions (19), are collected here. COLLECTION OF FORMULAE FOR USE IN COMPUTING 139 Collection of Formula r = line resistance per unit length, ohms. g = line leakage per unit length, mhos. L = line self-inductance per unit length, henrys. C = line electrostatic capacity per unit length, farads. / = frequency in cycles per second. I = line length in units used above. x = 2 wfL. Z = r+jx = z(6,. 1/ = tan- 1 -. y = -^— = -^- = Vg^+¥. fl^tan" 1 -. z = ^— = -^-= Vr 2 + r cos B z sin 6 S b = 2irfC. Y^g+jb^y/dj,. 9 & y = — - — = cos 0„ sin By V = VZY = v/6v = a+jp. v = Vzy. 20 V = 0, + V . a = v cos 0„. jS = v sin 0„. c/ = y| = M/0 Jf . u = \J-. 20 tt = 0*-0„. A = cosh VI = cosh al cos /3Z + j sinh al sin 01 = a/6 a . S = sinh VI = sinh a£ cos 01 + j* cosh al sin j8Z = s/0,. (Express cosh VI and sinh VI in the form of scalars with their angles in the same way as was done for the im- pedance Z and admittance Y above). B = U sinh VI = 6/06. b = us. 6 b = U + 0,. C = ^ sinh V7 = c/0 c . c = -. C = 0. - 0«. The values of the three constants A, B, and C, to be used in the general form of equation (31), are obtained by the above formulae. Of course, the symbol C as used in these equations is not to be confused with the same symbol when used to indicate electrostatic capacity. 140 ALTERNATING CURRENT PHENOMENA TABLE XVIII Tabular Form for Computing from Formula on Preceding Page r 6.560 log cosh al 0.16919 i o.iooxio-^ log sinh al 0.03588 0.003788 log COS lil +9.88128 C 0.00790 X 10-« log sin 01 -9.81223-10 f 800 cycles I 200 miles log cosh al cos 01 +0.05047 log2x 0.79818 log sinh al sin fil -9.84811-10 log/ 2.90309 log tan d a -9.79764-10 log L 7.57841-10 log cos 6a 9.92790-10 log 2 rf logC 3.70127 1.89763-10 A = S log a 0.12257 log X 1.27968 cosh VI \ 6a 327.890 deg. log r log tan d g log sin 6, log 2 e, 0.81690 0.46278 9.97564-10 1.30404 70.990 deg. log sinh al cos 01 log cosh al sin 01 log tan "., log sin 6, +9.91716-10 -9.98142-10 -0.06426 -9.87924-10 log b 5.59890-10 sinhWJ 10 ^ 0.10218 log fir 3.00000-10 310.777 deg. log tan By 2.59890 log sin By 0.00000 B = )log b 2.95475 log y 5.59890-10 U sinh VI \ 6b 301.344 deg. 0y 89.856 deg. C =jlogc 7.24961-10 log V 2 2 6 v 6.90294-10 160.846 deg. ^sinh V1 1 C 320.210 deg. e v 80.423 deg. log cos 6 V 9.22108-10 Numerical Res ults. Eauation31 log r 8.45147-10 log sin 6 V 9.99390-10 A 1.3261 /327.890deg. log a 7.67255-10 B 901.1 /301.344deg. log/3 8.44537-10 C 0.001777 j /320.210deg. log 360/2 t 1.75812 log 0° 0.20349 a 0.0047049 A minus sign before the logarithm 0° 1.59768 deg. means that the quantity represented at 0.94098 by the logarithm is negative. /3°Z 319.536 deg. log 2/y 5.70514 2 u -18.866 deg. "i log l 2.85257 -9.433 deg. The numerical values entered in the table correspond to a line whose constants are those given in the first six entries of the table, and are approximately those for No. 8 B. & S. gauge wire spaced 18 inches. See " Standard Handbook." Constants for loop of two wires. CHAPTER VII THE PROPAGATION CONSTANT, V = VZY = a + j(3 85. General. — In the preceding sections it has been shown how to obtain the numerical values of the quantity V = a -f j(i which enters into the exact solution for a line carrying an alter- nating current. In Chapter VI it is seen that a, the real portion of V, is the term which accounts for the decrease in the magnitude of the component waves in the direction of their propagation. For this reason the name "Attenuation Constant" has been given to the quantity a. The quantity j3, as the imaginary portion of V, determines the amount of rotation of the vectors representing the component waves for each unit of line length. /3 is numeri- cally the rotation of each vector, in radians, per unit length. Since a complete rotation of one of these component vectors takes place in a distance along the line equal to one wave length, the constant /?, which determines this distance, is termed the "Wave-length Constant." The combination of the two quantities into the single complex, V = a + j/3, is known as the "Propaga- tion Constant," since this quantity completely determines the manner of propagation along the line of the separate component waves, and thus their resultant. Both components of V are de- pendent upon all of the line properties — resistance, self-induc- tance, leakage, electrostatic capacity, and frequency. 86. Fundamental Equations for a and p. — The attenuation constant, a, depends, in the main, upon the line resistance and leakage, but its value is also affected by the other three properties. By equation (5), Chapter VI, a = y/\ (zy -xb + rg), (1) which, when expanded by x = wL, b = ojC, z = Vr 2 + x 2 , y = Vg 2 + 6 2 , becomes a = \/\ { V(r 2 + « 2 L 2 ) (g 2 + J&) + rg - o> 2 LC \ . (2) 141 142 THE PROPAGATION CONSTANT The quantity /3 depends principally upon the frequency, self- inductance, and capacity, though it is not independent of resist- ance and leakage. By equation (5), Chapter VI, = y/\ (zy + xb- rg), (3) which by the above expansion becomes $ = Vh WV^tfL?) (c? 2 + c^C 2 ) - rg + <**LC}. (4) The manner of variation of these two quantities, a and /3, with variation of the several separate quantities entering into their determination, has been extensively studied by Pupin, Fleming, Heaviside, and others, on account of their importance in deter- mining the operating characteristics of long electric cables. If / = 0, co = 0, and the expressions reduce to a] -V^. fi\ = 0, (5) Thus, V = y/rg + j = v as used in the discussion of direct J M =o current transmission over leaky lines, Chapter II. If r = and g — 0, we have immediately oL = 0; j8~L = Vxb = 2 irfVTC. (6) Jg=0 J ff =0 87. Wave Length and Velocity of Propagation. — In any case the distance along the line required for a complete rotation of the component vectors is 2x X = — = wave length, (7) P and the velocity of the separate waves is 2irf S = /X = —^- = velocity of propagation. (8) P Thus, with no line losses (a = 0), xl = 2 * = 1 —, J«=o 27I-/VLC fVLC' and a.-s] - Jcr=0 27T./VLC Vlc (9) SPECIAL CONDITIONS 143 The velocity of propagation when there are no line losses is 1/VlC. This is the limiting value of the velocity, and is equal to the velocity of light = 3 X 10 10 cm. per second, for conductors in air, when the internal self-inductance of the metallic conductors is neglected. Except under one special condition, the introduction of energy losses due to resistance of the conductors or leakage between the conductors, will cause the value of S to be smaller than S x , although, for all values of r and g the velocity S at continuously increasing frequencies approaches S^ as a limit. This approach to S^ at infinite frequencies arises from the preponderance of the zy and xb terms in equation (3), so that *L=vfe= s - (l0) Also, at high frequencies, the internal self-inductance of the con- ductors diminishes, and finally becomes zero, so that at the limit, the velocity, £~L =a0 , becomes equal to that of light because of L becoming equal to the external self-inductance only. Of course, at these very high frequencies, the radiation of energy into space would introduce losses and thereby increase the effective values of r and g. Since equation (10) was set up under the assumed condition that r and g remain constant, it is, of course, not true in general when r and g are both functions of the frequency. If the product rg increases less than in proportion to the square of the frequency and L and C remain constant, then equation (10) remains true. The subject of radiation losses cannot be taken up here. 88. Values of a, P, and S, when gL = rC. If equations (2) and (4) be expanded, «-ft lW + wW 2 + o>yL 2 + o> 4 L 2 C 2 )* + rg - <£LC }]*, = ft { W + <» 2 r 2 C 2 + «VL 2 + a> 4 L 2 C 2 )* - rg + co 2 LC |]*, (11) which, by adding and subtracting 2 ufrgLC to the term in paren- theses (see Fleming, "Propagation of Electric Currents," page 69), reduce to the following form: a = \/\ { V(gr + tfLCY + « 2 (gL - rC)* + (gr - rfLC) } , j 8 = V / i{V5r + co 2 LC) 2 + co 2 ^L-rC) l -(gr-co 2 LC) [.. (12) 144 THE PROPAGATION CONSTANT The form in (12) permits of a more ready determination of the manner of variation of the values of a and /3 than those previously given, but as far as computing purposes are concerned it is open to the same objections as equation (5). If we assume that the fundamental constants r, L, g, and C do not change with the frequency (an assumption sometimes far from the truth), the manner of variation of a and /3 with the frequency can easily be determined by inspection of equations (12). If the relation gL — rC — be fulfilled, the equations for a and /3 reduce to a\ = V7g, /sl = oVZC, (13) JjL-rC JffL-rC indicating that a is independent of the frequency and /3 is pro- portional to the frequency, a in this case is equal to the direct- current attenuation constant, and the velocity of propagation JgL=rC VLC is constant, irrespective of the frequency. If the relation gL — rC = be not fulfilled, the second term in parentheses under the double radical in the expression for a has a value, either positive or negative, and since its square is always positive, the value of a will continuously increase with increasing frequency — from its direct-current value, Vrgf, towards some limiting value at infinite frequency. jS 2 increases continuously with increasing frequency, from at zero frequency towards (o^LC + constant) at infinite frequency. 89. Infinite-frequency Values of a, P, and S. — The best way to establish the infinite-frequency values of a and /3 is to develop their equivalent expressions into power series in -, and then to note the finite terms when co = oo . Beginning with the fundamental equations (1) and (3), and letting - = w, r = mx, l = v > g = vb > ( 15 ) we have z = Vr 2 -!-* 2 = x (1 + u 2 )*, y = b (1+ v 2 )*, (16) , LOW-FREQUENCY VALUES 145 which, when expanded by the binomial theorem and multiplied, give zy = xb (1 + \u 2 + h v 2 - } u 4 + I uh 2 - £ v* + Aw 6 -tVwV- t VwV + 1 V« 6 + • • • )• (17) The fourth order terms in (17) may be combined into -\{u 2 -v 2 ) 2 and the remaining higher order terms in the series under the symbol <A (u, v), so that zy = xb [1 + h 2 + i«" - I (w 2 - » 2 ) 2 + iA (w, *)]. (18) If this value of zy be substituted in" equations (1) and (3), for a and /3, a = V| rg + \ xb [u 2 + v 2 - \{u 2 - v 2 ) 2 + 2^ (w, v)], , (19) /3 = V- i rgr + \ xb [4 + u 2 + v 2 - \ (w 2 - v 2 ) 2 + 2^ (u, v)} The terms x6[-i(w 2 -y 2 ) 2 + 2^(w,y)] vanish at infinite frequencies because they are all of at least the second order in -. Therefore, on substituting for the various 0} remaining quantities their expressions in terms of the fundamental line constants, (20) The curve for a, plotted in terms of the frequency, is asymptotic to the horizontal line at a height given by a] /=Q0 in (20). The curve representing /3 is asymptotic to the straight line passing through the origin, whose equation is /3 = 2 irj \ /r LC. As before mentioned, the velocity then approaches ■ r—^ as a limit. 90. Low-frequency Values of a, P, and S. — To investigate the forms of the curves at very low frequencies, the expressions for a and may be expanded in terms of o> instead of — . Omitting the 0} 146 THE PROPAGATION CONSTANT details, which are much the same as in the previous case, except that we let x = Mir, z = r (1 + Ui 2 )*, b = Vig, y = g (1 + t>i 2 )*, the following series are obtained: a = V7g VI + \ ( Ml 2 + v, 2 ) - tV (u x 2 - t>i 2 ) 2 + * * (ui, t>0 a =V r ^l+-_ + _- j, /8= Vi a* + i rg [u? + »i 2 - \ (ui 2 - »i 2 ) 2 + 2 * ( M| , „,)], (21) p = „y-LC + 5^— + ^ - B «^ -yj +. ... Thus the Umiting velocity of propagation at zero frequency is So = s] -§.- * (22) It is easily seen that the quantity in parentheses in (22) can never be less than 2 LC, so that the value of /3 is always greater than w VLC, except in the special case already mentioned, when gL = rC, for which the parenthesis becomes equal to 2 LC. In general, then, the velocity, S, is always less than the infinite frequency velocity. The two curves representing a and S as functions of the fre- quency start horizontally from their zero-frequency values, rise with increasing rapidity along curves approximately parabolic, reach their maximum slope at inflexion points where their second derivatives have a value of zero, and then, with continuously decreasing slopes, run out asymptotic to their infinite-frequency values. For fines with low resistance and leakage, the approxi- mately parabolic portions of the curves for low frequencies are very small in comparison with the large sweep of the curves in passing out to their final asymptotic values. With large losses, the curves flatten and approach their final limits more slowly. 91. Illustration of the Variation of a, (J, and S with Frequency for Power Circuit. — The following table of values has been computed, to show the manner of variation of these quantities for the power transmission line previously considered, and whose con- stants per unit length are again given at the beginning of the table. VARIATION OF a, ft AND S 147 TABLE XIX Different-frequency Values of the Attenuation Constant, a, Wave-length Constant, in Degrees, 0°, and the Velocity of Propagation, S, in Miles per Second, for L = 0.002040, r = 0.2750, g = 0.15 X 10r*, C = 0.01460 X 10- 6 / a f s 0.00020310 94,007 2 0.00022583 0.006888 104,530 4 0.00025820 0.012050 119,510 6 0.00028454 0.016401 131,700 8 0.00030483 0.020412 141,090 10 0.00032070 0.024253 148,440 12 0.00033317 0.028014 154,210 16 0.00035098 0.035456 162,450 25 0.00037205 0.052264 172,200 60 0.00039077 0.11942 180,870 180 0.00039528 0.35418 182,960 300 0.00039566 0.58973 183,140 420 0.00039577 0.82540 183,180 540 0.00039581 1.06111 183,200 660 0.00039583 1.29684 183,210 780 0.00039585 1.53259 183,220 900 0.00039586 1.76833 183,220 2000 0.00039587 3.9294 183,230 3000 0.00039587 5.8941 183,230 4000 0.00039588 7.8588 183,230 6000 0.00039588 11.7882 183,230 8000 0.00039588 15.7176 183,230 Inf. 0.00039588 Inf. 183,230 148 THE PROPAGATION CONSTANT V-f - j«_ 9 1 « k » VI v\ o ^ co a ■ 3_ c E ^! ~ ei © 2» a- | c -I g 8. I "3 CO 5 x CO ■ - -* CM o -I 3 RELATIONS BETWEEN a, 0, AND S 149 The lower-frequency values entered in Table XIX are plotted as curves in Figs. 49, 50, and 51, in order to show the curve shapes. From an inspection of the table, it is seen that at 60 cycles, a and S have nearly reached their limiting values, and /3 is approximately s x 10 17 16 15 14 13 12 10 9 Fn que ncy /, < Jjrcl esp erS BOO id. 8 i 4 6 8 T 1 i 1 i \ Fig. 51. — The Velocity of Propagation as a Function, of the Frequency. (Circuit Constants Representing a Power Line.) proportional to the frequency. The infinite-frequency values of a and S are nearly twice their zero-frequency values. 92. Relations between a, |3, and S. — The similarity between the curve representing a as a function of the frequency and that representing S is so striking that an explanation is needed. In equation (4a), Chapter VI, we have This may be written 2 a/3 = rb + gx. 2aP = w(rC + gL), (23) 150 THE PROPAGATION CONSTANT from which, in the expression for velocity, we have e _ u _ < ^ a ^~/3~ rC + gL' . rC + gL P m — o w - (24) In the first of equations (24), rC + gh is constant, so that the curves for a and S as functions of the frequency differ only by a constant factor. Thus, knowing any one of the three quantities, a, j8, or S, the other two may be quickly determined by means of equation (24) without recourse to the more complicated general expressions for the separate quantities. 93. Numerical Illustrations for Telephone Circuits. — To further illustrate the variation of a, /3, and S, the values entered in the following table were computed. The circuit constants used are approximately those for two No. 12, N. B. S. gauge copper wires spaced 12 inches — a standard type of aerial telephone circuit. The values of the constants are for a loop-mile, that is, wire to wire. TABLE XX The Propagation Constant and Velocity of Propagation in an Aerial Telephone Circuit of the Constants: r = 9.94, L = 0.00366, g = 1.5 XlO" 5 , C - . 00822 X I0r* f a 0° s 0.003861 0.00000 88,560 5 0.003874 0.02026 88,850 10 0.003909 0.04015 89,660 20 0.004035 0.07780 92,540 30 0.004203 0.1120 96,400 50 0.004573 0.1716 104,890 100 0.005381 0.2917 123,410 150 0.005967 0.3945 136,870 200 0.006392 0.4911 146,620 300 0.006942 0.6783 159,220 400 0.007260 0.8648 166,520 600 0.007582 1.2423 173,870 800 0.007725 1.6255 177,180 Inf. 0.007949 Inf. 182,310 The above quantities are shown graphically in Figs. 52, 53, 54, and 55. NUMERICAL ILLUSTRATIONS FOR TELEPHONE CIRCUITS 151 s s5 <T? ■>! o B 8 g u o 1—1 ►J. B n_ ft *i 1- >> a a a £ 5- o d +3 | Q o i 3 1 a i> ,c £ a O O 0) ,3 H H g " "? -a s 5 H o a> - " «J 1 s >> - § 0" f £ s s <j s • s 160,000 160,000 140,000 Zy 190,000 120,000 % 110,000 t 100,000 90,000 80,000 Fr< ■qu( ni'j ./. Cyc >s i eri |60( ml. i T L D 1&) 200 1 1 to 280 1 Fig. 54. — The Velocity of Propagation along a Telephone Line. a p 0.007 1.1 '/' 1.3 a % v • 0.006 1.2 V ' 1.1 J> '• / 0.005 1.0 »^ x- • X 0.0 -X- / 0.004 n> / 0.7 |^ / 0.003 o.n / / 0.5 ,/ / 0.002 o.t / 0.:i / 0.001 0.2 / / 0.1 / V / y F reqi lent y.f C>x les J'CT Soc< jnd Sx io-» 200 160 120 ■HO 40 160 320 480 640 «00 " Fig. 55. — The Attenuation Constant, Wave-length Constant, and Velocity 152 of Propagation for a Telephone Line. CHAPTER VIII THE EQUIVALENT IMPEDANCE OF CIRCUITS AND THE DETERMINATION OF THE FUNDAMENTAL CONSTANTS BY MEASUREMENTS 94. General Expression for Generator-end Impedance. — The complex ratio between the generator voltage and the genera- tor current, when these two quantities are expressed in complex form from the vector diagram representing them, is the equivalent impedance of the transmission line, including whatever load im- pedance may be placed at that end of the system. The expres- sion for this impedance is set up in the same manner as for the analogous direct-current problem, viz., by taking the ratio between the general equations in (19), Chapter VII, for E and I in terms of E Q and To- The load-end impedance is the ratio between the load voltage and the load current. Z = -j^ = load-end impedance. (1) By substituting, E = Zolo in equations (19), Chapter VII, and dividing, the common factor 7 canceling from both the numerator and the denominator, we have Z cosh VI +U sinh VI Z a = cosh VI + jj sinh VI E = y = equivalent generator-end impedance. (2) In treating the subject of line impedances, it has been the custom of some writers to introduce separate names for the various ratios of e.m.f. to current, and these names are not, in general, such as to be self-explanatory. The introduction of such terms as "Initial sending-end impedance," "Final sending-end impedance," and lastly — the least comprehensible of all — "Final receiving-end impedance," which is not, as would be 153 154 THE EQUIVALENT IMPEDANCE OF CIRCUITS supposed, the ratio between load voltage and load current, serve to confuse the mind of the reader. In this discussion we will confine ourselves, in general, to two symbols for equivalent impedance: Z = ratio between load voltage and load current, and Z g = ratio between generator voltage and generator current. The subscripts used are such as to indicate directly the meaning of the symbol. Both of the impedances, Z and Z g , are complex, of the form Zo = r + jxo, Zq — f g -\- JXg (3) 95. Generator-end Impedance under Special Conditions. — Two equations arise from (2) as special cases, which are perhaps more important than the general form. They are the expressions for Z g when the load end of the line is short-circuited, and when the load end of the line is open-circuited. In the first case, Z = 0, and in the second case, Z = oo . Introducing these values of Zo and evaluating the resulting equations, Zg] = Z g ' = Ut&nh VI, .ta-o = short-circuited impedance. (4) Zg] = Zg" = £/cothW. = open-circuited impedance. (5) These two equations are analogous to equations (39) and (40), in Chapter II, to which they would reduce for / = 0. If, for the purpose of analysis, we assume that the line losses are zero (r = and g = 0), the equations (4) and (5) assume trigonometric forms because of a being zero. Thus, since tanhj/3 = jtan/3, and cothj/S = — jcot/3, (6) Z '\ =jU tan /3Z, Ja=0 Z g f> ] = -jU cot pi Ja-0 Again, when a = 0, the quantity U is no longer complex, but "L-vl-vl-v/1- <?> GENERATOR-END IMPEDANCE 155 and by equation (6), Chapter VII, 1 = co VIC, Ja=0 so, finally, the equivalent generator-end impedances of a line with no losses, the load end being short-circuited and open-circuited, are respectively, zA =jy^ tan co VLCZ, Z/'l = -3\k cot uVZCl. (8) In such a system it is apparent that, for given values of L and C per unit length, doubling the line length at constant frequency will produce the same variation in Z g ' and Z g " as doubling the frequency with the original line length unaltered. This comes about from the fact that co and I enter the equations (8) as a product, and in no other way. The product VLCl may also be written VLtCt, where L t and C t represent respectively the values of the total line self-inductance and the total line capacity. In (8) the resulting impedances are either pure inductive or pure condensive reactance — no real component appearing in the equations. This must be obvious from physical considerations. If a real component appeared in the expressions for impedance, an energy loss would thereby be represented, which is contrary to the assumption that there is neither resistance nor leakage present. It is quite obvious that at low frequencies the short-circuited impedance of a line with zero losses will be inductive reactance and approximately equal to coL*, numerically. The first of equa- tions (8) reduces to this for low frequencies by neglecting all terms in the series expansion of higher order than the first in co. It is also obvious that for an open-circuited line at low frequencies the impedance will be condensive reactance and approximately equal in magnitude to -7=- . This is the first term in the series expansion coC* of equation (8), for Z". With increasing frequencies, neither of the two quantities, self-inductance and electrostatic capacity, can be considered alone in such a simple way, but their combined effect must be summed up in equations of the form of those in (8). 156 THE EQUIVALENT IMPEDANCE OF CIRCUITS 96. Analogies between Circuits with Distributed and with Lumped Properties. — It is instructive to compare the results given by (8) for the impedance of circuits having distributed constants with the forms of expression for circuits with lumped constants, which, in an approximate manner, represent physically the short-circuited and open-circuited transmission line. In the case of a short-circuited line we may consider that the current flowing from the generator has, in effect, two alternative paths — one around the entire loop of the line, and thus through the entire amount of self-inductance, and the other through the line electro- static capacity. The phenomena in this case are approximated by those which take place in a parallel circuit, one path of which contains a condenser and the other an inductance coil. In the £ Short-oiixuited Liue L t C { £ Open-circuited Line Analogs. y W v Fig. 56. — Simple Circuits Analogous to Short-circuited and Open-circuited Transmission Lines. case of an open-circuited line, the analogy is different. Here the generator current may be supposed to flow through two parts of a circuit in series — a portion of the line constituting a self -in- ductive impedance and the electrostatic capacity of the two wires completing the circuit by a condensive impedance. If the capacity and self-inductance in the localized system be made equal, re- spectively, to the total capacity and self-inductance of the trans- mission line, as shown in Fig. 56, the curves giving the reactances of the different circuits appear as in Figs. 57 and 58, which are plotted with frequency as the independent variable for the 400 mile line whose values of L and C are given in Table XII. Thus L t = 0.816 henry and C t = 5.84 X 10"* farads. ANALOGIES BETWEEN CIRCUITS 157 X .400 1 j / L200 at / L000 800 l» 1 600 1 1 400 1 1 a/ m / /a — ■ 200 /> Fi equenc vJ 4 f |L .'0 It •0 a fei ^2 o_ _JB *? —a B— 1-360— --400 ■400 \< s> ■"* . • Ron / A ' J / ■800 / / •000 i i / •200 (» a / L400 / i -L600 1 Fig. 57. — The Equivalent Impedance of a Short-circuited Line and of Its Analogous Circuit. 1 x 1400 A X L200 <•/ y^ -• L000 S ''' 800 .</ K 600 ,s' 400 '7 200 /' c/ t f F •('(JV cue r4 -200 1 A t )^> **\ D 1 iO a : !40 % <0 8 J^- ' 8 D 400 ■400 / /v Qj -000 i 1 / 800 It 1 1000 II 1200 c/ 1400 r 1000 \ Fig. 58. — The Equivalent Impedance of an Open-circuited Line and of Its Analogous Circuit. 158 THE EQUIVALENT IMPEDANCE OF CIRCUITS The formulae for the simple analogous circuits are: 1 uLt (Short-circuited line) Z = . _ .1 ' 1 - u*L t Ct J " C <- J ^L t (Open-circuited line) Z = j I wL« -^ J • (9) Numerically, the equations for the actual lines and the analo- gous circuits are: 1. Short-circuited Line (a) Line equation, zA = j 373.8 tan 0.013716/. Ja = 5 127/ (b) Analogue, Z = j ± , a000J8813/2 - 2. Open-circuited Line (c) Line equation, Z a "\ = -j 373.8 cot 0.013716/. Ja-0 (d) Analogue, Z = j (s.127/ - aQ000 3 5858/ ) The lettering of the curves in Figs. 57 and 58 corresponds to that in the above numerical equations. In both comparisons, the two curves in question coincide at zero frequency, but for frequencies above 20 cycles per second they separate rapidly. For the line, the first resonance frequency, as indicated by the infinite value of impedance for' the short-circuited line and by the zero impedance for the open-circuited condition, is 114.5 cycles per second. The 400 mile line is exactly one- quarter wave length at this frequency. This value of / may be determined by placing equation (a) equal to infinity. Then tan 0.013716/= inf., or 0.013716/= (2»-l)|, /«fe^f, /= (2n- 1)114.5. The analogous lumped circuits reach their resonant condition at a much lower frequency than the transmission system — in this case at a frequency of 72.8 cycles per second. The ratio between these ANALOGIES BETWEEN CIRCUITS 159 two frequencies, 114.5 and 72.8, is =, and this ratio holds true for any length of line, as may be shown in the following way. The first resonance frequency for the line occurs at a quarter wave length, or when fil = ~ . The resonance frequency in the lumped circuit occurs when the inductive reactance is equal to the con- densive reactance, numerically. Thus, for the line at the first resonance condition 2 4 VL t Ct and for the lumped circuit 2rfL t = ^, f = 2*fC t ' J 2-K^L£ t The ratio of line frequency to lumped-circuit frequency, at reso- . - 2i 7r nance, is thus -r- = -• 4 2 In both Fig. 57 and Fig. 58, it is seen that the lumped-circuit impedance is equal to the line impedance at frequencies other than zero — that is, for the frequencies at which the respective curves intersect. In this case, at approximately 200 cycles, the imped- ances are again equal for both the short-circuited and open- circuited conditions. The values of the frequencies for equal values of impedance may be determined by equating the respective expressions. For the short-circuited line and its analogous circuit, from equations (8) and (9), for equal impedance, \/^tan c VLA = , . «fr , (10) and for the open-circuited line, from the same equations, 5-.von--i.-fc 01) w coC< Equation (11) reduces to (10) by taking the reciprocal of both members and then multiplying throughout by y^, which shows w that the impedances in the respective cases become equal at the same frequencies. These transcendental equations, (10) and (11), 160 THE EQUIVALENT IMPEDANCE OF CIRCUITS are incapable of direct solution for u, which must, therefore, be determined by methods of approximation or by taking the inter- section of curves. The equations and curves for the equivalent impedances of a line with zero losses and with distributed self-inductance and capacity are of importance in determining the free frequency of oscillation of such circuits, and, in this connection, will be referred to again. 97. Limiting Conditions at Infinite Frequencies. — In actual lines, resistance and some leakage are always present, so that the ideal conditions just considered are never realized exactly. In the actual case, the resulting impedance does not consist of reactance only, which reaches alternately infinite and zero values according to tangent or cotangent trigonometric functions. The energy losses in the line require a real component in the impedance expressions for short-circuited and open-circuited lines, and prevent the reactance curves from reaching infinite values, though they may pass through zero values. The real component of the resulting impedance can, of course, never be negative, for this would imply an average flow of power from the line into the genera- tor, and such a condition is impossible, if the line be either short- circuited or open-circuited as assumed. However they may vary, then, the angles of the two quantities Z g ' = U tanh VI and Z " = U coth VI can never atyain the limiting values +90° and —90° no matter how long a line or how high a frequency be considered. Since the attenuation constant, a, approaches a finite limit at very high frequencies, and the wave length constant, /3, approaches proportionality with the frequency, it may be shown that, at very high frequencies with a line of fixed length, the curves representing the equivalent resistance and equivalent reactance, for both Z ' and Z g ", approach strict periodicity, with frequency as the inde- pendent variable. At very high frequencies, the quantity U — y y. approaches the constant y ^ in value. The imaginary portion of U disappears on account of the preponderance of the reactance and susceptance over the resistance and conductance, respectively. By equations (20), Chapter VII, at infinite frequency, a = a x = c onstant, = o>VLC, INFINITE-LINE IMPEDANCE 161 so that and finally, zj\ - y ^tanh (la* +j2rfVLCl), Zff "] ." vi coth **- + j 2irf ^^ (12) At these frequencies, the real portion of VI is constant, and the imaginary portion proportional to /, so that the resulting im- pedances as given by (12) are periodic in respect to the frequency, with a period of /o = . — = / , which is the number of V LCI vLtCt cycles increase in frequency required to bring about a recurrence of the same values of impedance. It will be remembered that, in the case of a line with no losses, a = 0, the impedances were both periodic in I with / constant, as well as periodic in / with I constant; and further, periodic in the product fl. The impedance of a line with losses is periodic in / at infinite frequencies only, while with fixed frequency it is never strictly periodic in length as independent variable. If the line length be increased, both components of VI increase proportionately, so that, though the resulting curves showing the equivalent resist- ance and equivalent reactance, when plotted in terms of line length, present some of the characteristics of a periodic function, they are not strictly periodic since they do not present recurrently the same sequence of numerical values. 98. Infinite-line Impedance. — If the line be infinitely long, then VI becomes infinity, and since both the hyperbolic tangent and cotangent approach unity for infinite values of argument (the argument may be either real or complex), the values of open- circuit and short-circuit impedance approach each other for increasing line length, and finally become "-VI - infinite-line impedance. (13) If the attenuation constant a be zero, in which case V is a pure imaginary, the impedances do not approach U in value with in- creasing line length; for, in such an event, the hyperbolic tangent and cotangent reduce to the trigonometric tangent and cotangent 162 THE EQUIVALENT IMPEDANCE OF CIRCUITS respectively (neglecting multiplication by ±.j), so that for I = oo the expressions for impedance become indeterminate. In no physical line is it possible for a to be absolutely zero, so that the impedances always approach U = infinite-line impedance, as a limit. Equation (2), the general expression for equivalent impedance when the line is neither short-circuited nor open-circuited, also reduces to U for infinite values of I — that is, the equivalent generator-end impedance in such a case is independent of the load- end impedance, Z . 99. Numerical Illustrations of Equivalent Line Impedances. — For the power transmission line, whose constants are given in Tables XII and XIII, the following numerical values of open- circuited and short-circuited impedances are obtained, by equations (4) and (5) for a line length of 100 miles. TABLE XXI Short-circuit and Open-circuit Impedances op a 100 Mile Line at Ddtferent Frequencies r = 0.275, L = 0.00204, y = 0.15X10-», C = 0.0146 X10- 8 / Impedance in ohms Short-circuited Open-circuited 27.5 +j0 66,680.0 -jO 25 27.6+J32.1 293.2 -j'4331.1 60 28.3+J77.8 58.7 -j 1789. 8 180 36.8+J264.5 15.2-;'526.7 300 75.8+J616.4 12.7-j'225.2 420 865.7+;' 2593. 9 13.9 -j49. 2 540 165.3 -j 1272.0 18.0 +j 107.5 660 29.5-J449.4 29.6+J309.0 780 16.2-J188.2 82.0+ j 735.4 900 14.6 -j20. 8 3,255.5 +./4428. 1 The curves in Figs. 59 and 60 are plotted from the above values. In this case, where the line losses are very small, the equivalent resistance and reactance reach enormous values. For this line, quarter-wave-length phenomena appear at a frequency of 458.06 cycles per second, and half-wave-length phenomena at 916.12 cycles per second. In the short-circuited condition, the impedance NUMERICAL ILLUSTRATIONS OF LINE IMPEDANCES 163 rtj \ 1 [600 r m) 71 L200 000 / / I 800 too ' f rl 400 •m I V x^ pf ■qlt anov. f a; =-- K a 10 8 K) 400 500 a 7f >0 800^- 400 | •^CC 800 ' 1200 x, 1 Fig. 59. — The Equivalent Impedance of a Short-circuited, 100 Mile Power Line. X r '7 700 70 600 GO 500 50 V 400 40 /' 300 :so 200 20 100 10 r ^ n Pre iue icy / -100 1( a X) a Q 400/ 1 500 8 X) K a 8 X) 900 -200 -300 /a -400 •500 -600 -700 Fig. 60. — The Equivalent Impedance of an Open-circuited, 100 Mile Power Line. 164 THE EQUIVALENT IMPEDANCE OF CIRCUITS reaches very large values at the frequency which makes the line a quarter-wave length. Beginning at zero frequency, the reactance increases approximately along the tangent curve shown in Fig. 57, but, instead of reversing sign through infinity, reaches a finite maximum value at a frequency slightly less than 458 cycles, drops rapidly through zero and to a negative maximum at a frequency slightly greater than 458 cycles. The numerical values of this negative, or condensive, reactance decrease, then, along a curve approximating the tangent curve, until at a frequency of 916 cycles the half-wave-length phenomena appear and the reactance passes through zero. With continuously increasing frequency, above this value, the curves are sensibly periodic, for, from Table XIX, a has become practically constant. It is interesting to note that, at the frequency corresponding to half-wave-length phenom- ena, the effective generator-end resistance is only a little more than one-half its zero-frequency value. The open-circuited impedance is, of course, very high at low frequencies. The equivalent resistance at zero frequency is 66,680 ohms — slightly greater than the reciprocal of the total line conductance, 10 6 100 X 0.150 = 66>6670hms - The equivalent reactance at zero frequency is zero, but it rises rapidly to very high numerical values, approximating the recip- rocal of the total line susceptance. The numerical value of this negative, condensive, reactance decreases along a curve approxi- mating the cotangent curve shown in Fig. 58, passes through zero at a frequency of 458 cycles per second, where the quarter-wave- length phenomena appear, and then continues approximately periodic in frequency as independent variable. At a frequency of 300 cycles, the equivalent resistance is less than one-half the total line resistance. When longer lines or lines with larger values of r and g are considered, the equivalent resistance and reactance do not pass through such extended ranges of values. As a comparison with the above case, the values of the short-circuited and open-circuited impedances of the No. 12 N. B. S. gauge telephone circuit, for which the values of a and /3 are given in Table XX, are tabulated. The length of line is taken as 300 miles. DETERMINATION OF LINE CONSTANTS 165 TABLE XXII Short-circuited and Open-circuited Impedances of 300 Mile Telephone Line r = 9.94, L = 0.00366, g = 1.5X10-*, C = 0.00822X10^ / Impedance Short-circuited Open-circuited 5 10 20 30 50 100 150 200 300 400 500 600 700 800 2112 -yo 2108 -y 80 2095 -j 158 2047 -y 307 1973 -y 437 1776 -y 627 1291 -j'745 1003 -y 642 865 -y 514 791 -y 345 773 -y 297 720 -y 263 698 -y 200 713 -; 177 693 -y 176 3137 -yo 3074 -; 374 2902 -y 689 2423 -y 1047 1991 -y 1125 1478 -y 989 1093 -j 651 995 -y 513 936 -y 461 808 -y 397 732 -j 297 732 -y 230 722 -y 221 686 -; 189 692 -y 148 As may be seen by an inspection of the values of impedance in Table XXII, a line of this length and with these values for the fundamental constants, has no inductive reactance at low fre- quencies even when short-circuited. For the range of frequencies covered in Figs. 61 and 62, plotted from the above table, the reactance is condensive throughout — both when short-circuited and when open-circuited. At very high frequencies, the equiva- lent reactances decrease, numerically, and finally oscillate about a zero value, while the equivalent resistances oscillate about some finite limiting value. A comparison of Figs. 61 and 62 with Figs. 59 and 60 shows strikingly the effect of an increase in the values of line resistance and leakage, and therewith an increase in the attenuation con- stant a. 100. Determination of Line Constants from Short-circuit and Open-circuit Tests. — It is frequently necessary to determine, by actual measurement, the effective values of the four funda- 166 THE EQUIVALENT IMPEDANCE OF CIRCUITS m r m MOO /a> The eft e •tivc v.- tl MM MM in rm iOOO \ 1 rni &■ " uencica ihowo p 100 po V BOO 200 ^r 200 800 ;• _ 100 (00 Fre |UC ncy ft \Vul J 1 itS I't'O ad f 100 1 i 300 400 , " T 800 Fig. 61. — The Equivalent Impedance of a Short-circuited, 300 Mile Telephone Line. X ■1120 r 2S0O ^ ■1040 ••960 21011 I COI le effective i idenslve|thrc 'eactaDCt uriiuut 13 lie •■880 ran geo I In que icie -Shr VTI1. ■800 20(H) V' •720 ••640 1600 •560 ■•480 1200 •■400 A ■320 800 aj •240 r -160 400 S**" -80 R en enc y.f Cyc M 1 >erf C'CO n<l. m ) too 2 • 300 400 500 600 71 ') i ID Fig. 62. — The Equivalent Impedance of an Open-circuited, 300 Mile Telephone Line. DETERMINATION OF LINE CONSTANTS 167 mental line constants at some particular frequency. On account of the variation of these constants with changes of frequency, it is, in general, not possible to measure the resistance and leakage of the line with direct current, nor to measure the capacity and self- mductance by any ballistic galvanometer method. If alternating current be used to make such determinations, all of the line con- stants come into play, so that any one measurement will not serve to determine the four constants. The frequency used in the measurement must be the same as that for which the values of r, L, g, and C are desired. Of course, in some cases, where approxi- mate results only are desired, such simple procedures as the deter- mination of the line capacity from an open-circuit test by the equation Ct = ttTW farads, where E g and I g are respectively the generator voltage and gen- erator current when the line is open-circuited, may suffice. How- ever, if the line is electrically long — that is, approaches a quarter wave-length — or if the resistance or leakage coefficients are large, such a procedure may not yield even an approximate result. Accurate values of the constants may be obtained by using two measurements — one of the short-circuit impedance and one of the open-circuit impedance. When both components of the imped- ances are measured, four numerical values are obtained as data from which the values of the four line constants may be calculated. Suppose that, by measurement at the particular frequency for which the effective values of the fine constants are desired, actual numerical values are obtained for and Zg = short-circuit impedance Z g " = open-circuit impedance, the line length, I, being known. The unknown quantities are Z and Y per unit length, or U and V, which are expressed in terms of Z and Y by equations (4) and (6), Chapter VI, and from which Z and Y may be obtained. Now Z ' = UttmhVl and Z " = UcothVl, 168 THE EQUIVALENT IMPEDANCE OF CIRCUITS so that by multiplication U = VZ g 'Z g ". Further, sinh VI e w - <-n _ 1 - t~™ so that l- € -2Vl (14) tanh VI = cosh VI Z.'» I 1 + e ,-2 VI' (15) which, on solving for e 2 n = ryf|i gi yes 2Vl _ U + Z g ' ~U-Z a '' or v = h ln zr^6 -nw+m. (16) where the notation, In x, signifies "the logarithm to base c of x." U 4- Z ' The quantity 77 -^-, is complex, of the form A -\- jB f and it U Zlg is of this quantity that the logarithm is required. Since e «*+>" = c « (cos v +j sin v) = e u /v (radians) , we have u-\-jv = In e u (cos v + j sin v) = In (e u [v). Therefore, In (A + jB) = ln(VA 2 + B 2 /4) = \nVA 2 + B 2 +jd, where Thus B 6 = arc tan -r- A V = ~ln VA 2 + B* +j ± arc tan | With C/ and V from the preceding equations, U = sJy and V = VzF, Z = r+jaL = UV and which thus determines the four line constants. (17) (18) Y = g+ju>C = $j, (19) METHODS OF MAKING IMPEDANCE MEASUREMENTS 169 101. Multiple Solutions. — In using equation (18) it is im- possible to avoid a certain ambiguity. In taking the arc tan -j, r A any multiple of 2 ir may be added to the result at will, thus per- mitting of a variety of values for the imaginary portion of V. Multiplying (18) by Z, Vl = al+ j(3l = ^ In VA 2 + B 2 + j i arc tan j- al = 1 In VA 2 + B 2 , (31 = ■= arc tan -j- (20) When the line length and frequency are not too great and the conductors are in air, an estimate as to the probable value of /3Z may be made in order to decide whether or not a multiple of 2 w is to be added to arc tan -j. In (20), 01, itself, may differ from the true value by any multiple of ir, since one-half the arc tan -j is involved. If the measurement be made upon a circuit consisting of a long cable, where it would be impossible to estimate accurately the number of wave lengths involved, it would be necessary to commence the measurements at frequencies for which the cable would certainly be less than one-half wave length, and then, by taking measurements at a number of increased frequencies, to plot the impedance curves in order to decide upon the number of times ir must be added to ^ arc tan -j in determining &l by (20). The quantity a is obtained without any difficulties of this kind. When the line is very long, the differences between the measured imped- ances and U will be very small, since U is the infinite-line im- pedance, so that considerable inaccuracy may result from equation (16) on account of the small difference between two nearly equal quantities in the denominator. 102. Methods of Making Impedance Measurements. — The impedance measurements may be made by the voltmeter-ammeter- wattmeter method, or by means of an alternating current bridge, depending upon the amount of power available and upon the character of the transmission line under investigation. When the 170 THE EQUIVALENT IMPEDANCE OF CIRCUITS first method is used, in the case of a power line with large con- ductors, it will probably be difficult to obtain the real component of the impedance accurately on account of the very low power- factors — particularly with the line open-circuited. When an alter- nating current bridge is used, the current and voltage impressed upon the line will probably be much smaller in value than those obtaining when the line is in actual use. However, as long as no corona formation occurs when the line is in use for its regular purpose, and no ironclad circuits are present, the values of the constants obtained by a bridge method with small amounts of power would not be subject to change when greater voltage and current are used. The voltmeter-ammeter-wattmeter method of measuring both components of impedances is so familiar as not to need description. 103. An Impedance Bridge. — A form of alternating current bridge which the writer has found to be very satisfactory is shown in Fig. 63. In the two upper figures, a and b, the bridges are shown dia- grammatically, and in the lower figure are shown the connections to a double-throw, double-pole switch to be used for changing from scheme a to scheme b. R h R 2 , and R are adjustable non- inductive resistances, and C is an adjustable standard capacity. Connection "a" gives directly the two components of a condensive admittance, Y = g + juC, so that it is necessary to take the reciprocal of this value to obtain the impedance. Connection "b" gives the two components of an inductive impedance, Z — r + jb)L. The formula? to be used in the reductions are given beneath the diagrams, and in them the values of R, R h R 2 , and Co which produce a balanced condition in the bridge are to be substituted. To detect a condition of balance, either a telephone receiver or a vibration galvanometer may be used — the latter being, in many instances, the more satisfactory. When a telephone receiver is used, frequencies much below 250 cycles per second cannot be employed with the bridge on account of the inaudibility of their tone, and again, with any circuit as unknown whose apparent constants change with the frequency, a balance at one frequency will not be a balance at another frequency, so that with any ordinary source of e.m.f. difficulty will arise from the presence of higher harmonics in the e.m.f. wave. The sound produced by them will always be present and serve to mask the disappearance PRECAUTIONS IN THE USE OF THE BRIDGE 171 of the fundamental tone at a condition of balance. The vibration galvanometer, on the other hand, may be used at low frequencies as well as at frequencies of hundreds of cycles per second, and since the period of the instrument is adjusted to coincide with the period of the e.m.f. used, no trouble will arise from the presence of higher harmonics of ordinary magnitudes — say a few per cent. Connection a. " R Rj c=£c Connection 6. r _ R» R2 L = R Co <\j C AA/V\AA K^> AA/WNA Unknown Fig. 63. — An Impedance Bridge. For Ri and R 2 , resistances with several steps and with a total value of about 2000 ohms each are appropriate, except for the measurement of small inductances, by connection "b," where values as low as 50 or 100 ohms may profitably be employed. 104. Precautions in the Use of the Bridge. — To avoid excessive values of R it is often desirable to shunt the unknown with a known non-inductive resistance when measuring condensive admittance by connection "a," or to insert a known non-inductive resistance in series with the unknown when measuring inductive 172 THE EQUIVALENT IMPEDANCE OF CIRCUITS reactance by connection " b," The values of g or r thus introduced are to be subtracted from the measurements. The alternating e.m.f. supplied to the bridge should be taken from the secondary of an air-core or other insulating transformer placed at a distance sufficient to avoid the influence of extraneous magnetic fields. This insulating transformer is needed to prevent the introduction of errors on account of the large electrostatic capacity between the generator windings and leads and earth. If great accuracy is desired, the different portions of the bridge should be electro- statically shielded, and, in any case, care should be taken so to distribute the different parts of the apparatus that the mutual interactions be reduced to a minimum. Considerable difficulty is often experienced in finding the proper adjustment for a balance. When measurements of a transmission line are undertaken, one usually has no knowledge of the approxi- mate values, even, of the unknowns, so that a balance must be sought by trial. When a long line is involved and the frequency is high, it cannot be known beforehand whether the unknown im- pedance is inductive or condensive. Trial must then be made of both bridge connections. Since there are two quantities which require simultaneous adjustment, C and R, the best that can be done is to first adjust one of them until the current in the gal- vanometer is a minimum, then adjust the other until a new mini- mum is obtained, and so on. A little experience enables these adjustments to be made very rapidly after a condition is found where a change in either C or R produces a noticeable change in the galvanometer current. Care must be taken to see that the frequency used does not approach those for which the impedances under measurement vary rapidly, for then small variations in the frequency of the e.m.f. supplied to the bridge would cause the bridge settings for a balance to vary greatly, and thus prevent any accurate adjustments. For example, in an impedance measure- ment of the short-circuited 100 mile line, whose effective resistance and reactance are shown as a function of the frequency in Fig. 59, reliable measurements would be difficult with frequencies between 350 and 500 cycles per second. According to Fig. 60, for this same line open-circuited, measurements would be difficult with fre- quencies between 800 and 1000 cycles per second, as well as at 450 cycles, where the impedance would change from inductive to condensive, or vice versa, for small changes of frequency. EFFECT OF ERRORS IN ORIGINAL DATA 173 105. Illustrative Computation. — To illustrate the use of equations (14) to (19), consider the following numerical example. On a given line, 41 miles in length, impedance measurements with the following results were made at a frequency of 660 cycles per second : Line short-circuited. Connection " b " Line open-circuited. Connection " a " ft = 1000 Ri = 1000 R = 1224.0 Co = 0.2766X10"* Ri = 1000 R 2 = 1000 R =816.7 Co = 0.7816X10-* By using the respective formulae in the two cases, "a" and "b": Short-circuit test, "b," Z ' = 817 +i 1147 ohms. Open-circuit test, "a," Y g " - 0.0012244 +j 0.0032411 mhos. 1 Y " 1 o = Z g " = 102.0 - j 270.0 ohms. These values are to be used as original data with which to enter equation (14). The computations are most easily done with logarithms, according to the following table, where in place of the quantity itself, its logarithm is given, unless otherwise stated. <f> represents, in general, the angle of a complex quantity; for example, Z = r +jx = z[$. These are the effective values of the line constants at this fre- quency (660 cycles per second) per mile of length, and they will, in general, be different for different frequencies. 106. Effect of Errors in Original Data. — In this numerical example, the fundamental data, Z g ' and Z g ", are such as to yield very accurate results, for the two impedances as measured are sufficiently different from each other to permit an accurate for- mation of their sums and differences, as required in equation (16). Further, the resulting values of impedance and admittance per unit length of line have angles which are not very nearly equal to either or 90 degrees. The determination of the quantity g is usually the most unsatisfactory, for the angle of the admittance, Y, is frequently very near 90 degrees, so that g, the real component, 174 THE EQUIVALENT IMPEDANCE OF CIRCUITS TABLE XXIII Calculation of Link Constants from Two Impedance Measure- ments. Equations 14 to 20 EJ 2.91222+ j 3.05956 tan <t> 0.14734 sin <f>' 9.91089 Z " 2.00860 -j 2. 43136 tan <t>" -0.42276 sin*" - 9.97103 z„' 3.14867/54.538 deg. *'" „ 2. 46033/ -69. 305 deg. . 5. 60900/ -14. 767 deg. COS * u 9.99638-10 V 2. 80450/ -7. 3835 deg. sin 4> u -9.10893-10 U 2.80088 -j 1.91343 u 632.25 -J81.93 t Numericallv 817.00 +j 1147.00 f Numerically *; Let m = U + Z ' ' and n = U - Z„' m=U + Z„' n=U-Z a ' 1449.25 +j 1065.07 ) M -184.75 -j 1228.93 J w umerically m 3. 16114 +j 3. 02738 tan <f>m 9.86624-10 COS <j>m 9.90623-10 n 2.26659 -j 3. 08953 tan <f> n -0.82294 sin* n - 9.99515 m 3.25491 /36.313 deg. n 3.09438 /261.450 deg. A +JB = - 0.1605 3/134.863 deg. (Logarithm of scalar) log log - 9.20557-10 <f>° = tan- 1 j 134.863 deg. log Mod. 9.63778-10 log* 2.12989 log In — n 9.56779-10 , 360 log 2T 1.75812 log 2 J 1.91381 log arc tan -r 0.37177 log 2 1 1.91381 s'm<t> v 7.65398-10 +j 8. 45796-10 tan <f> v 0.80398 9.99471-10 V 8.46325-10/81.075 deg. U 2.80450 -7.383 deg. 4>z cos 4> t z sin <t> z r X 73.692 deg. 9.44841-10 1.26775 9.98216-10 0.71616 1.24991 3.61772 7.63219-10 4>v cos 4> v y sin 4> v I 2 c f 88.458 deg. 8.42985-10 5.65875-10 9.99984-10 4.08860-10 5.65859-10 3.61772 2.04087-10 Final numerical results r = 5.2018, L = 0.0042873 , = 1.2263Xl(H C = 010987 X lO-* EFFECT OF ERRORS IN ORIGINAL DATA 175 is not accurately determined. In the particular case just treated, the angle of Y is 88.458 degrees — only 1.542 degrees different from 90 degrees. In general, the error in the angle of Y is of a lower order of magnitude than the original errors in the measured angles of the equivalent impedances, Z g and Z g ", so that in this case, the cosine of the angle 88.458 degrees, on which the determi- nation of g depends, is still obtained with considerable accuracy. To illustrate the effects of an error in the original measurements, suppose that, in determining Z g ', an error of one degree is made by a wrong determination of the imaginary component of the impedance — the real component being assumed correct. Then as original data, in logarithms, Z g ' = 2.91222 + j 3.07570 = 3.15951 /55.538 deg. Z " = same as before. From this fundamental data, r = 5.0394, L = 0.0043677, g = 1.4026 X 10"*, C = 0.010874 X 10- 6 . In this latter case, the angle of Y is 88.219 degrees, as compared with 88.458 degrees in the previous case based on the correct values of Z ' and Z g '\ The change in the resulting angle of Y due to a change of one degree in the angle of Z g ' is 0.239 degrees. As mentioned before, the best results from measurements of this kind are obtained under conditions which cause the two impedances, Z g and Z e " } to be considerably different, for then the value of the fraction in equation (16) is obtained without loss of accuracy. The shorter the line under test, the greater the differ- ence in these impedances. With very short lines, however, when the resistance and leakage coefficients are small, it will be difficult to obtain accurate values of the real components of the effective impedances by measurement, so that, though the subsequent numerical solution will be easy to carry out accurately, the errors in the fundamental data may be serious. With the 300 mile tele- phone circuit, whose impedances are tabulated in Table XXII, for example, values of line constants based on impedance measure- ments at frequencies above 100 cycles would be quite inaccurate, on account of the very nearly equal values of Z g ' and Z g ". If the effective line constants were desired at higher frequencies, it would 176 THE EQUIVALENT IMPEDANCE OF CIRCUITS be desirable, or even necessary, to make the impedance measure- ments on a shorter section of the line — say 100 or 50 miles, depending on the frequency. On the other hand, measurements on a 300 mile power transmission line, whose values of r and g are usually much smaller than for the telephone circuit, would prob- ably furnish very satisfactory data. CHAPTER IX CHANGE OF WAVE SHAPE ALONG LINES 107. Method of Treatment of Complex Wave Shapes. — The equations given in the foregoing chapter are all based on the complex quantity representation of alternating quantities, and thus yield solutions for a single frequency. By the definition of an ideal line r however, in which it is stated that the effective values of the line constants in such an ideal line are independent of the magnitudes of the voltage or current, the treatment of complex wave shapes by the superposition of separate solutions for the separate frequencies is made possible. If the e.m.f. wave of a generator which supplies a particular line contains a number of higher harmonics, the voltage wave shape at any point in the line may be found by determining the magnitude and phase position of the several component voltages at the point in question by applying the complex quantity method of analysis to each sepa- rately, and then taking the instantaneous sums as data from which to plot the resultant curve — if the curve, and not merely the values of the separate components, be needed. The method of treatment is essentially the same as used in the correction of oscillographic curves of wave shape for the effect of distortion circuits, Chapter IV. If any of the generator e.m.f. harmonics should coincide in frequency with a resonance frequency of the line, and the line losses be small, very great exaggeration of this particular harmonic, either in the voltage or in the current pro- duced thereby, may take place in portions of the line. For a con- sideration of such phenomena, it will be necessary to reproduce, in the form for use in alternating current systems, some of the specialized equations given in Chapter II for leaky direct current lines. 108. General Case ; Load Impedance = Z . — The funda- mental equations are: E = E- cosh VI + hU sinh VI, I = 7 cosh VI + E A sinh VI, 177 (1) 178 CHANGE OF WAVE SHAPE ALONG LINES which give the voltage and current at any point distant I from the load-end of the line in terms of the load-end voltage and current. Solving simultaneously for E and h (see eq. 16, Chap. II), E = E cosh VI - IUsinh VI, I = IcoshVl-EjjeinhVl, (2) which give the voltage and current at any point distant I from the generator in terms of the generator voltage and current, E and /. To determine the voltage and current at any point in a given line when the load-end impedance, Z , is known, let k = total line length between generator and load, Z , and let E' and /' represent the voltage and current respectively at any point distant V from the generator, whose voltage is E. V is thus always less than lo. The ratio between current and voltage at the generator is the equivalent generator-end impedance, which, by equation (2), Chapter VIII, is Z cosh Vk+U sinh VI* Z a = Y a = cosh Vlo + jj sinh V k 1 and / = EY a (3) Therefore, applying equation (2), for a distance I' from the genera- tor and eliminating I by equation (3), E' = E (cosh VV - UY sinh VI'), r ' = e( Y cosh VV -jj sinh ')1 (4) In computing from this equation, first determine the value of Y g and then substitute it in the equation in connection with the particular values of V. No benefit is to be derived from the substitution of the expression in (3) for Y g in equation (4), except numerically, as mentioned. 109. Short-circuited and Open-circuited Line. — Since con- siderable labor is required to determine the values of Y„, in general, the two special cases — short-circuited line and open -circuited line — are" of importance, for then the values of Z g or Y g are easily SHORT-CIRCUITED AND OPEN-CIRCUITED LINE 179 obtained. For these special cases, the substitution of the expres- sions for the equivalent generator-end impedance from equations (4) and (5), Chapter VIII, gives Short-circuited : sinh V(k-l') E'~\ = E (cosh VI' - coth Vk sinh VI') = E Jz„=o /'] = 5 (coth Vk cosh VI' - sinh VI') = E Jz =o U sinh Vlo coshV(lo-l') U sinh Vk (5) Open-circuited : E'~\ = E (cosh VI' - tanh Vl sinh VI') = E Jz =oo V\ = ^ (tanh Vk cosh 7Z' - sinh VI') = E- _|Z„=oo C coshF(Zo-Q cosh F/o sinhF(Z -ZQ [/ cosh Vlo (6) If we make V = Zo, then equation (4) must reduce to forms similar to equations (19) and (20), Chapter II — giving the load voltage and current in terms of the load impedance, generator voltage, and total line length. Making the substitution, V = k, and reducing, F % cosh Vk +Y Usmh Vlo' r g Zo cosh Vk + U sinh Vk From these, for the two special cases, Eo\ =EsechVk, JZ =<x I (7) r = E -p: cosech Vk- Jz =o U (8) In a direct current system, the quantity V is always real, and thus in equation (8), for instance, the quantity E sech Vk con- tinuously decreases for increasing line length. In the alternating current system, V is complex, and a function of the frequency, so that the value of sech Vk may vary through very great numerical ranges, and in the case of a line with no losses in which the attenua- tion constant is zero, throughout the range from — oo to +«>. Similarly with cosech Vl - The voltages at the terminal of an open-circuited line may thus rise to very high values, as well as 180 CHANGE OF WAVE SHAPE ALONG LINES the current at the end of a short-circuited line. Equations (5) and (6) indicate that excessive voltages or currents may also occur at intermediate points in the line length, if the total line length, k, be such as to give large values to coth Vk or to tanh Vk, depend- ing upon whether the line is short-circuited or open-circuited. If the line be connected to an impedance Z , the excessive currents or voltages may or may not occur, depending on the resulting value of Y g , from equation (3). Y g becomes very large when Z cosh Vlo+ U sinh Vk approaches zero. For example, in a line with no cosh Vk sinh Vk U losses (a = = cosjSZo, = j sin Pk, 0), so that Y g = oo when Z cos fik + j y -x sin &k = 0, or when 'o = -JVc tan ^- Infinite values of Y g , however, imply an equivalent short- circuit on the generator supplying the line, and under such a condition it is not reasonable to suppose the generator voltage to remain constant, for then we would have infinite currents through- out the line, as well as infinite voltages. In such cases — very large values of Y g — the phenomena must be expressed in terms of the generator current J, rather than the generator voltage, E, the former being more or less definite at short-circuit, while the latter disappears. The equations in terms of I are obtained by substituting IZ g for E, and then, from equations (4), (5), (6), (7), and (8), we have E' = I (Z g cosh VV - U sinh VV), r = I (cosh W - jf sinh Vl'\ E'\ _ = IU(tsLBhVkcoshVV-8ii^Vn=IU sm ^y~ V) , /'] = I (cosh W - tanh Vk sinh VV) = I coshy (*»-*'> Jz =o coshyfo (9) (10) NUMERICAL ILLUSTRATION 181 coshF(Z -O 1 W =IU(cothVl Q coshVl'-smhVl')=IU /'I = I (cosh W - coth Vk sinh VV) - 7 - smh 7 ( *° ~ -^o = i sinh VL (11) Y cosh F^o + yy sinh Vl Io = cosh VIq + yr sinh FZ o = IU cosech FZo, JZ =oo o = 7 sech FZo- (12) (13) These equations will serve to determine the voltage at any point in a line, under any condition, and will also furnish the basis for a discussion of the phenomena of quarter-wave transmisson. Consider first the distortion of wave shape of voltage in an open- circuited line when the generator voltage contains higher har- monics of small magnitude in comparison with the fundamental. The wave shape of the generator voltage then differs little from a sine curve. Of course, if no higher harmonics are present in the original wave shape of the generator, there will be no distortion. Referring to equation (6), suppose the line length to be one- seventh of a quarter wave length for the fundamental, or approxi- mately, j3i°Zo = 13 degrees. When the line losses are small, the voltage of fundamental frequency does not change very much throughout the entire line length. The third and fifth harmonics of voltage are considerably magnified, while for the seventh har- monic, tanh Vk becomes very large, infinity if a be zero, so that this particular harmonic attains values relatively much greater than either the third or the fifth. It will thus greatly alter the shape of the e.m.f. wave at points along the line, as well as at the load end. A numerical example serves best to illustrate this effect. 110. Numerical Illustration. 100 Mile, Open-circuited Line. — The 100 mile transmission line whose fundamental constants are r = 0.275, L = 0.00204, g = 0.15 X 10" 6 , and C = 0.0146 X 10" 6 , 182 CHANGE OF WAVE SHAPE ALONG LINES has, for the frequencies which are the odd multiples of 60 cycles per second, and which are the only ones which would generally be present in the e.ra.f. wave of an alternator, the propagation- constants and open- and short-circuit impedances given in Tables XIX and XXI. In this case lo = 100 miles. The coefficients of E in equations (6) may be termed the voltage and current coefficients, respectively. Numerical values of these coefficients for different frequencies and different distances V, from the generator are given below in Table XXIV. TABLE XXIV Factors for Obtaining the Distortion of Voltage and Current Wave Shapes along a 100 Mile Line when Open-circuited Or (a) Voltage coefficients, -=■ , as scalar with angle / /' = 25 i' = 50 J' = 75 1 ' = 100 60 1.009 - 0°.21 1.016 - 0°.36 1.020 - 0°.44 1.021 - 0°.47 180 1.097 - .76 1.168 - 1 .25 1.211 - 1 .52 1.226 - 1 .61 300 1.387 - 2 .11 1.684 - 3. 12 1.871 - 3 .61 1.934 - 3 .76 420 3.476 - 13 .63 5.539 - 15 .81 6.895 - 16 .60 7.367 - 16 .81 540 0.654 -163 .03 2.145 -170 .69 3.192 -171 .92 3.567 -172 .20 660 0.203 - 10 .38 0.665 -174 .86 1.319 -176 .91 1.563 -177 .27 780 0.473 - 2 .51 0.260 -174 .10 0.878 -178 .41 1.119 -178 .86 900 0.678 - 1 .72 0.340 -144 .28 0.717 -179 .32 1.001 -179 .87 (6) Current coefficients, -= X 10* JS , as scalar with angle f r = o r = 25 i' = 50 V = 75 60 0.558 88°. 12 0.420 88°. 05 0.281 88°. 00 0.140 87°. 98 180 1.898 88 .35 1.464 88 .14 0.996 87 .99 0.504 87 .90 300 4.434 86 .77 3.608 86 .38 2.546 86 .12 1.316 85 .98 420 19.545 74 .24 17.385 73 .63 12.998 73 .25 6.944 73 .04 540 9.173 -80 .50 9.387 -81 .46 7.627 -82 .00 4.262 -82 .29 660 3.222 -84 .53 4.149 -86 .19 3.785 -86 .94 2.242 -87 .30 780 1.351 -83 .64 2.714 -87 .34 2.912 -88 .40 1.855 -88 .85 900 0.182 -53 .68 1.971 -87 .68 2.676 -89 .28 1.867 -89 .83 Suppose that the generator supplying this line has an e.m.f. wave shape whose equation is, as a function of 6 = 2 irft, e = 100,000 sin d + 8000 sin (3 0-40°) + 6000 sin (5 0-295°) + 5000 sin (7 0- 110°) + 2000 sin (9 0-50°) + 500 sin (11 0-170°). (14) EQUATIONS FOR INSTANTANEOUS VOLTAGES 183 The equation for any particular component of the e.m.f. at any point in the line is obtained by multiplying the equation for the component of the same frequency in e, by the factor, or voltage coefficient in part (a) of Table XXIV, and then changing the phase angle as given in (14) by the angle of the factor. Thus, at a distance of 75 miles from the generator, the maximum value of the ninth harmonic in the voltage wave is 3.192 X 2000 volts = 6384 volts, and its equation as a function of time (or 0) is e 9 | = 6384 sin (9 - 221°.92). J/'=75 111. Equations for Instantaneous Voltages. — The following are the equations for the voltages at distances of 25, 50, 75, and 100 miles from the generator. 25 miles from the generator, V = 25, e' = 100,900 sin (0 - 0°.21) +8776 sin (3 - 40°.76) +8322 sin (5 0-297°. 11) + 17,380 sin (7 0- 123°.63) + 1308 sin (90- 213°.03) + 102 sin (11 - 180°.38). (15) 50 miles from the generator, V = 50, e' = lOl,6OOsin(0-O°.36) +9344sin(3 0-41°.25) + 10,104 sin (5 0-298°. 12) +27,700 sin (70- 125°.81) +4290 sin (9 - 220°.69) +332 sin (11 - 344°.86). (16) 75 miles from the generator, V = 75, e' = 102,000 sin (0 - 0°.44)' +9688sin(3 0-41°.52) + 11,226 sin (50 - 298°.61) +34,480 sin (70- 126.°60) +6384sin(90-221.°92) +660 sin (11 - 346°.91). (17) 184 CHANGE OF WAVE SHAPE ALONG LINES 100 miles from the generator, V = 100, at open end, e' = lO2,lOOsin(0-O°.47) +98O8sin(30-41°.61) + ll,6O4sin(50-298°.76) +36,835 sin (70 - 126°.81) +7134 sin (9 - 222°.20) +782 sin (11 - 347°.27). (18) 112. Equations for Instantaneous Currents. — Using the current coefficients from part (6), Table XXIV, the following equations are obtained for the line currents at the several different distances from the generator. Obviously, the current is zero at the open end of the line, where V = Iq = 100. Generator current, at V — 0, t = 55,8OOsin(0 + 88°.12) + 15,184 sin (3 + 48°.35) +26,604 sin (5 - 208°.23) +97,725 sin (7 - 35°.76) + 18,346 sin (90- 130°.50) + 1611 sin (11 - 254°.53). - (19) 25 miles from the generator, V = 25, i' =42,000 sin (0 + 88°.05) + 11,712 sin (3 + 48°.14)' +21,648 sin (5 - 208°.62) +86,925 sin (7 - 36°.37) + 18,774 sin (90- 131°.46) +2074 sin (11 - 256°.19). (20) 50 miles from the generator, V = 50, i' = 28,100 sin (0 + 88°.00) +7968 sin (3 + 47°.99) + 15,276 sin (5 - 208°.88) +64,990 sin (7 - 36°.75) + 15,254 sin (90 - 132°.00) + 1892 sin (11 - 256°.94). (21) 75 miles from the generator, V = 75, t' = 14,060 sin (0 + 87°.98) +4032 sin (3 + 47°.90) +7896sin(50-2O9.°O2) +34,720 sin (7 - 36°.96) +8524 sin (90- 132°.29) + 1 121 sin (1 1 - 257°.30) . (22) DISCUSSIONS OF WAVE SHAPES 185 Fig. 64. — Voltage Wave Shapes along a 100 Mile Open-circuited Power Line. 113. Discussions of Wave Shapes. — The curves representing the foregoing equations are shown in Figs. 64 and 64a. In both figures, the upper curve represents the generator voltage wave shape. These curves were drawn by means of the curve-tracer ISO CHANGE OF WAVE SHAPE ALONG LINES Generator Voltage Equation 11 Generator Current Equation It Current at i •»» Equation 20 Current at J'— 50 Equation 21 Current at I '=75 Equation 22 Fig. 64a. — Current Wave Shapes along a 100 Mile Open-circuited Power Line. mentioned in section 42, Chapter III, and, though they may be slightly inaccurate, serve sufficiently well to show the char- acteristics of the different wave shapes. In the particular line to which the foregoing curves apply, the DISCUSSIONS OF WAVE SHAPES 187 total length of circuit is 100 miles, and therefore approximately equal to one-quarter of the wave length for the seventh harmonic — 109 miles, from the value of (3° given in Table XIX. The voltage and current coefficients for this frequency attain values much larger than for any of ( the other frequencies, and therefore the comparatively small seventh harmonic in the generator voltage produces very prominent voltages and currents of this frequency throughout the line. Although the seventh harmonic in the generator voltage is only five per cent of the funda- mental, the seventh harmonic in the voltage at the free end of the line is thirty-six per cent of the fundamental. In the capacity, or charging, current of the line, the seventh harmonic component is approximately twice as great as the fundamental. The maximum instantaneous values of voltage at different points along the line are indicated on the respective curves. On account of the different phase positions of the several component harmonics, the maximum value of the generator voltage wave is only a little greater than the maximum value of the fundamental — 104,000 maximum, with 100,000 volts as maximum value of the fundamental. At the open end of the line, the maximum value of the voltage wave is 148,000 volts, which, as an increase of nearly fifty per cent over the generator voltage, might be serious in its effect upon the line insulation. In a line designed to operate normally without corona formation at the voltage assigned to the fundamental frequency in this illustrative example, the presence of such a large superimposed harmonic which increases the maxi- mum voltage by nearly fifty per cent might be sufficient to start corona formation, and introduce a power loss then supplied, in part, by the fundamental frequency. Very considerable values of leakage, or corona loss, are required to materially decrease the voltage coefficients and thus to decrease the exaggeration of those harmonics for which the line approximates a quarter wave length. The value of g used in the computation of the data for this particu- lar case is 0.15 X 10 -6 , which corresponds to a power loss due to leakage, of 1500 watts per mile of each wire for an e.m.f. of 100,000 volts between wire and neutral. For a line differing, as in this case, by 9 miles from a quarter wave length for the seventh harmonic, this value of g is not sufficient to appreciably lower the value of the voltage coefficient in Table XXIV. 188 CHANGE OF WAVE SHAPE ALONG LINES 114. Dependence of Wave Shapes upon Generator and Load Characteristics. — The large value of the seventh harmonic in the generator current would seriously increase the generator heating, and demands, even under conditions of no-load, con- siderable generator capacity. The seventh harmonic current at the generator is 74.2 degrees ahead of the corresponding component in the terminal voltage, so that, for a considerable range of current values, whatever current of this frequency flowed through the inductive impedance of the generator windings would produce an increase of the terminal voltage (of this frequency) above the open- circuit value. No reliance could be placed on an assumption that in such a case the generator terminal voltage of this frequency would decrease greatly when the current flows. For a line a little longer than a quarter wave length, the current would lag by nearly 90 degrees, though the numerical value of the voltage coefficient might be the same as before. In such a case, the terminal voltage of the generator undoubtedly would decrease materially as soon as the current flowed. An increase of only nine per cent in the frequency (from 60 to 65 cycles for the fundamental) would cause the line to be an exact quarter wave length for the seventh harmonic, in which event, much greater distortion of the wave shapes would occur. A still further increase of the frequency would cause the line length to be greater than a quarter wave length for the seventh harmonic, and, depending upon the character of the source of e.m.f. of this fre- quency, might entirely alter the phenomena. The presence of a load at the terminal of such a line would also change the phenomena. If the load consisted of an inert imped- ance, Z , the distribution of electrical quantities along the line could be determined by equations (4), although not without rather tedious computations. If the load consists of a synchro- nous motor, for example, instead of an inert impedance, it is neces- sary to know the wave shape of counter electromotive force of this machine, and furthermore, to know the relative phase positions of its component harmonics of e.m.f. with respect to those of the generator supplying the line. The counter e.m.f. of fundamental frequency of the motor will be nearly in opposition to the e.m.f. impressed on the motor terminals, E , but, by reference to Fig. 33 it is seen that the phase angle between the generator terminal voltage, E, and the load voltage, E Q , may differ by large amounts, ENERGY SUPPLY THROUGH TERMINAL IMPEDANCES 189 depending upon the magnitude of the current I Q as well as upon the load power-factor. The possibility of a variation of, say, 60 degrees in the phase difference between E and E 0) allows the higher harmonics in the counter e.m.f. of the motor to occupy almost any phase position whatsoever with respect to the same harmonic frequencies in the generator e.m.f. If a particular harmonic is present in the generator wave shape, but is not present in the counter e.m.f. of the motor, then the motor constitutes practically a short-circuit at the load end of the line for this particular frequency. Similarly, the generator con- stitutes a short-circuit at its end for any harmonic present in the motor counter e.m.f. which is not present in the generator wave. The discussion of the mode of operation of the various machines which may be connected to a line must be taken up in another place; this section is intended to deal only with the calculation of the values of the voltage and current along the line when the terminal conditions are given. If both the generator and the load produce e.m.f. 's, as in the case of a higher harmonic in both the generator and motor ter- minal e.m.f. 's, the distribution of voltage and current along the line must be determined by equations identical with (55) and (56), or (57) and (58), of Chapter II; in which, to conform to the nota- tion used here for alternating current systems, the complex Zi z 2 Ei Ei Line Length = l e'„ Ii Zj Fig. 65. — Long Line Supplied through Impedances by Two Generators. quantities, U and V, must be substituted for z and v, respectively. Ei and E 2 are then the values (in the correct complex form to take account of their relative phase positions) of the voltages impressed on the two ends of the line, and E and I' and /" are respectively the values of voltage and current at a distance li from Ei and k from Ei. 115. Double Source of Energy Supply Through Terminal Impedances. — If impedances are placed between the sources of e.m.f. and the line, the circuit appears as in Fig. 65, and the solu- 190 CHANGE OF WAVE SHAPE ALONG LINES tion must be obtained as below. In this system, select arbitrarily as positive the direction indicated by the arrow. Then, by Kirchhoff's Law, Ei = E\ — IiZ\, hiz = ivji — E%. By equations (2), (23) E t ' = Ex' cosh Vk - Uh sinh Vk, h = h cosh Vk-jj Et' sinh Vk, (24) or, substituting (23) in (24) and combining terms, 7i [Zi cosh Vk + U sinh Vk] + ZJ* = E x cosh Vk + E 2 , 1 1 cosh Vk + jj sinh Vk — h = Ex jj sinh Vk, (25) from which, by solving simultaneously for h and h, Ik-. /* = \-^8intiVk\ + E t (Zi + Z 2 ) cosh Vk + (u + ^A sinh Wo Ei + # 2 (cosh Vk + ^ sinh 7Zo) (Zx + Z 2 ) cosh FZo + (u + ^f- 2 ) sinh Wo (26) Using the value of I\ from equations (26), the voltage and current at any point in the line are then given by E = Ei' cosh VI - hU sinh VI, I = h cosh VI - Ei' jj sinh VI, (27) where I is the distance of the point in question from the e.m.f. Ex, and E x ' - Ex - SJ> It is only rarely that conditions in any transmission system would be definitely enough known to warrant the use of equations (26) and (27). Since all of the quantities entering these equations are complex, actual numerical computations from them are quite tedious, though not difficult. APPROXIMATIONS BASED ON NEGLECT OF LINE LOSSES 191 116. Approximations Based on the Neglect of Line Losses. — In many cases it is not as important to know the actual values of the different harmonic voltages or currents along the line, as it is to be able to determine quickly the approximate maximum values which these quantities may attain. The presence of resistance and leakage in the line, leading to energy loss, serves always to prevent the attainment of infinite values of current and voltage, and, in general, serves to decrease the magnification of any particular harmonic, due to resonance conditions. Therefore, calculations made for a line with zero resistance and leakage, but with the same values of L and C as in an actual system, will yield values for the current and voltage distortion greater than could ever take place in the actual system, and thus serve as a guide in determining whether or not it would be necessary to make a rigid solution. Neglecting, then, the losses, r-#«j2*/ j8 = 2tt/VlC, and U -% so that, by cosh j(ft = cos (ft, and sinh j(ft = j sin {ft, equations (5) to (13) become, sin (k - V) '1 - Jz„=o sin $o cos g (jo - V) sinjSZo (28) E' V = E cos |8 (k - V) z =oo cos /3Zo -•jgi/jj fMft-*) ="» V J, JZ { cos /3J (29) Eq = h = E cos /% + j y 7y Y sin /SZo E Z cos /3Zo + j y p sin pk (30) 192 CHANGE OF WAVE SHAPE ALONG LINES 'sinpjlo- V) Jz =o " C cos/3/o j,l =/ co8<3(?o-n J z -o cos pk (31) E' r Z =<x> _ ., . /Z COS ft (*q - Q 17 Vc sinftio = 7 JZo«°° £ o =r 7o = sin /3(^> — Q sinftio (32) lc Y cos Pk + j\j sin /3Z : — 7c — cos |3Zo + jZ y j sin /% (33) Z?0 /o /o = E sec /SZo, Jz =» ]= —jEK/t cosec/3Z . Z =0 » -tv = 7 sec jSZo- = —j? V ^ cosec Pt<>> jz =o (34) (35) For either short-circuited or open-circuited lines, computation by the foregoing approximate equations is very easy and rapid. It is not proper, of course, to use them when they reduce to either infinite or zero values, for then the line resistance and leakage become the predominating quantities which control the distribu- tion of line voltage and current throughout. In the simple cases of short-circuited and open-circuited lines, the equations may be used except for values of /3Z approximating multiples of 90 degrees. In this event, since for power lines a is approximately constant for frequencies above 60 cycles per second, the magnitudes of the phenomena at these critical frequencies may be easily determined from equations (5) to (13) by the simplified expressions, COMPARISON OF APPROXIMATE AND EXACT SOLUTIONS 193 cosh ( al + j -z ) = j sinh al. COsh (al -\~jir) = — cosh al. cosh ial -\-j -tt-J = — jsinhaZ. sinh lal + j ~) = j cosh a£. sinh (al + j 7r) = — sinh aZ. }■ (36) sinh lal + j -~- j = — j cosh aZ. tanh (al+j-zj = coth aZ. tanh (al + j x) = tanh «Z. tanh f «Z + j —) = coth al. The above formulae are directly applicable to the discussion of quarter-wave transmission, where the frequency of the generator is adjusted to such a value that the line becomes a quarter wave- length. 117. Numerical Comparison of Approximate and Exact Solu- tions. — Applying the approximations, equations (29), to the line whose voltage and current coefficients for the different fre- quencies are given in Table XXIV, the following numerical results are obtained, which, as approximations, may be compared with the accurate values given in Table XXIV. In computing the following, the value of 2 irf VLC is used for 0, and not the true values, dependent upon the line resistance and leakage as given in Table XIX. In the following table, the complex quantity representing the designated ratios is given in the form of a scalar and its angle. In all cases, except for a frequency of 420 cycles per second, for which the line is nearly a quarter wave length, the approximations are sufficiently close to the true results for all, or at least most, engineering purposes, and the labor required to secure the approxi- mate results is practically negligible in comparison with that required by the rigid expressions. For this line, by equations (13) and (36) when the frequency is slightly higher than 420 cycles per second, so that 0k is exactly x , 194 CHANGE OF WAVE SHAPE ALONG LINES the voltage at the open-circuited load end is 25.26 times the genera- tor voltage. TABLE XXV Approximate Values op Voltage and Current Coefficients from Equation (29) for a Distance ({') of 75 Miles from the Generator. Total line length = 100 Miles. True values of coefficients taken from Table XXIV for comparison. E' E' /' /' / E , true -£-, approx. E 10*. true E 1V ' approx. 60 1.020 - 0°.44 1.020 0° 0.140 87°. 98 0.141 90° 180 1.211 - 1 .52 1.212 0.504 87 .90 0.504 90 300 1.871 - 3 .61 1.874 1.316 85 .98 1.319 90 420 6.895 - 16 .60 9.627 6.944 73 .04 9.943 90 540 3.192 -171 .92 3.228 180 4.262 -82 .29 4.310 -90 660 1.319 -176 .91 1.322 180 2.242 -87 .30 2.247 -90 780 0.878 -178 .41 0.879 180 1.855 -88 .85 1.857 -90 900 0.717 -179 .32 0.718 180 1.867 -89 .83 1.868 -90 118. Resonance Frequencies. — In the general case, with the line closed through impedances at both ends, the frequencies at which resonance phenomena occur may be easily determined from equation (26). When resonance phenomena occur, the line cur- rents become very large, even for small impressed electromotive forces, and in the limiting case, when neither resistance nor leakage is present, and consequently no losses, the currents become infinite with finite voltages impressed. The only way in which these infinite values of Ii and I 2 can occur is for the denominator in equation (26) to become zero. That is, the values of / which cause the denominator of this equation to reduce to zero, when the line losses are assumed zero, are those at which the resonance phenomena occur. Thus we have for resonance, from (26), (Z, + Z 2 ) cos 0k + j (v | + V i ZiZ,) sin A = 0. (37) Zi and Z 2 are functions of the frequency, and though they may be made up of any combinations of inductances and capacities, no conductance or resistance should be present if the above equation is to apply rigorously. /3 a^so is a function of the frequency, /S = 2x/VXC. The solution of equation (37) for / must be RESONANCE FREQUENCIES 195 obtained by approximation — a graphical process similar to that used in determining the free frequencies of vibration for the oscillograph vibrator, Chapter V, Fig. 21, being convenient. To illustrate the use of this equation in determining the critical, or resonance, frequencies of a line connected as indicated in Fig. 65, consider again the same 100 mile power transmission circuit. Let one end be closed through an inductance of 0.050 henry, and the other through a condenser of 0.020 microfarad and an in- ductance of 0.10 henry in series. Then Zi=j27r/X 0.050, Z 2 =i(2./x0.10-^_). (38) For the line, per mile, L = 0.00204 and C = 0.0146 X 10"*. Substituting these numerical values in (37) and expressing the result in such form as to utilize the tangent, we have, finally, tan0°.031269co = 1 50 X 10 6 - 0.15 co 2 co 7061.8 -0.000018376 co 2 ' (39) If we let y' represent the first and y" the second member of this equation, the intersections of the curves y' and y" plotted as functions of co will occur at those values of co which satisfy the above equation and which therefore are those corresponding to resonance conditions, or frequencies. These curves are shown in Fig. 66, from which intersections at the following values of co are noted: u, for resonance g>/2 x = frequency 2,302 366 7,200 1150 12,400 1970 17,600 2800 19,700 3130 23,800 3780 29,300 4660 33,000 5250 196 CHANGE OF WAVE SHAPE ALONG LINES Only the first two of these frequencies are sufficiently low to fall within the range of the higher harmonics produced by alternating current generators. It is interesting to note that the first reso- nance frequency, 366 cycles per second, is considerably lower than that for the open-circuited line alone, when /SZ becomes ~, or 90 degrees, at a frequency of 458 cycles per second, as 1 1 1 3.0 1 1 1 1 1 2.5 V 1 1 1 \ 1 f II 1 2.0 \ ,i 1 1 1 l 1.5 v' y '\ v'l ' 1 i V' I t 1 i ! i i 1 J 1.0 » J 1 1 1 ! \y" / / / r / ! 1 1 0.5 1 1 / » / ! V J / 1 1 1 1 / , i f v' 1 1 f / f / r / / <0 sid ■3/ 1 1 1 1 / 8 / / 12 0/ ■ / M V 32 i k / / / / / / \i ii / / ' 1 -~ Fig. 66. — Graphical Determination of Resonance Frequencies from Equation (39). Even with this comparatively short line — 100 miles — and the small value of electrostatic capacity placed at one end of the circuit, this first resonance frequency of 366 cycles per second is only the sixth harmonic in a 60 cycle system. An increase in the capacity and self-inductance present would cause resonance at a still lower frequency, and since the fifth harmonic is quite prevalent in generator voltage wave shapes, resonance phenomena of this frequency may well appear in a one hundred mile transmission system. For line lengths between 200 and 300 miles, the first resonance frequency appears at about the third harmonic of 60 cycles, or 180 cycles per second, so that resonance phenomena of this frequency may be produced if any source of e.m.f. of this frequency is present. RESONANCE FREQUENCIES 197 From equation (37), used in determining the resonance fre- quencies in this particular case, it is apparent that it makes no difference whether the terminal impedances are inserted with Zi at the generator end and Z 2 at the load end, or vice versa. Res- onance phenomena appear at the same frequency in both cases, since Zi and Z 2 enter equation (37) only as sums or products. Even if the magnitudes of voltage and current produced by resonance condition for one of these higher harmonics should not be such as to endanger the power system, or even to be observed in their effect on normal operating conditions, they still may be of sufficient magnitude to create serious disturbance in communi- cating circuits which parallel the power system. For this reason, particularly, it becomes desirable, or necessary in some cases, to eliminate, as far as may be, the possibility of abnormal values of voltage or current of these higher frequencies in power systems. The large magnitudes of voltage and current of fundamental frequency are not so serious in their effect upon communication circuits as the smaller voltages and currents of the higher fre- quencies. CHAPTER X POWER RELATIONS IN LONG LINES 119. Instantaneous and Average Power. — The instantaneous value of power at any point in a transmission line is equal to the product of the instantaneous values of voltage and current. In Fig. 31 and section 70 of Chapter VI is given an illustration of the instantaneous voltage, current, and power distribution along a line. The average value of power throughout a cycle is of more interest and importance. Explicit expressions (equations (3), (8), (18), (19), etc., Chapter VI) have been given for the voltage and current at any point in a line, and from these the equation for average power may be de- rived by taking the product of the scalar values of these quantities times the cosine of the angle of phase difference between them, P = ei cos <f>. By the relation that the cosine of the angle between two vectors is equal to the sum of the products of the direction-cosines of the two vectors, referred to two axes at right angles, a more convenient form of expression may be secured. Thus, referring to the e.m.f. and current represented analytically by E = d + jez and I = i x + fa, we have P = average power = ei cos <j> = e\i\ + e^h. (1) 120. The Separate Components of E and / at any Point. — Obviously, then, the problem of setting up an explicit expression for the average power at any point in a line resolves itself into obtaining expressions for the separate components of E and I at any point and then forming the sum of the products of in-phase components. The details of such a development are quite simple, but nevertheless tedious. 198 THE POWER EQUATION 199 The quantities entering into the expressions for voltage and cur rent are V = a+j(3, U = Sj y = u/6 u = wi + juz, 1 [Y ui .Wl . U = \-Z = ^- J u^ = yi + m- (2) (3) U 1 L W - U At the load end, let E = e f + to" and / = V + ji ". Then by equation (19), Chapter VI, the voltage and current at any point in the line distant I from the load end are given by E = (eo + to") (cosh al cos pi + j sinh al sin (il) + (*»' + jV) ( w i + 7U») (sinh ai cos /3Z + j cosh aZ sin /3Z), 1 = (i ' -|- fa") (cosh aZ cos /3Z + j sinh aZ sin fil) + («b' + to") (Vi + J2/2) (sinh al cos 0Z ■+■ j cosh al sin #). These quantities are of the form E = ei+ to and / = i x + jt2, and by multiplying in equation (3), we have ei = e</ cosh aZ cos 01 — e " sinh al sin /3Z + (Vwi — io'vv) sinh a? cos {il — (io'ih + to"wi) cosh al sin j8Z, e2 = e " cosh aZ cos /3Z + eo' sinh aZ sin (31 + (*o"wi + io'ik) sinh aZ cos /3Z + (io'ui — io'vv) cosh al sin /3Z. (4) t'i = tV cosh aZ cos 01 — to" sinh aZ sin /3Z + (eo'2/1 — eo"y2) sinh aZ cos jSZ — (eo'2/2 + eo"!/!) cosh al sin 0Z, t2 = t'o" cosh al cos /3Z + V sinh aZ sin /3Z + (eo"t/i + eo't/2) sinh al cos /3Z + (eo'yi — e "y2) cosh aZ sin /3Z. (5) 121. The Power Equation. — Substituting the above expres- sions for the separate components in the equation P = 01*1 + Ste, ( 6 ) 200 POWER RELATIONS IN LONG LINES multiplying out, and finally combining terms, there results the following form: P = ( eo V + eo'%") (cosh 2 al cos 2 01 + sinh 2 al sin 2 01) + [(eo' 2 + eo" 2 ) yi + (to' 2 + V 2 ) u,] cosh al sinh aZ - [(eo' 2 + eo" 2 ) v, + (V 2 + to" 2 ) u*] cos 01 sin /3Z + [(eoV + eo"to") (u lVl + u*y 2 ) + (eoV - eo'V) (Mit/j — M2?/i)] (cosh 2 aZ sin 2 01 + sinh 2 a£ cos 2 01). (7) Considerable care is required in reducing the product as indicated by equation (6) to the form given in (7) on account of the large number of terms involved. The power at the load end, or receiver, is Po = e V + e "to" and the scalar values of voltage and current at the load end are e^ = eo'2 + ^ and tf = V 2 + to" 8 . Introducing these values into equation (7), P = Po (cosh 2 al cos 2 01 + sinh 2 al sin 2 01) + (eo^i + t'o^i) cosh al sinh aZ — (eo 2 ^ + to 2 ^) cos 01 sin /5Z + [-Po (miVi + M22/2) + (eo'to" - eo'V) (M12/2 — M22/1) ] (cosh 2 al sin 2 /3Z -f sinh 2 aZ cos 2 /3Z) . (8) 122. Reduction in Form of the Power Equation. — Still fur- ther reduction is possible by a consideration of the necessary relations existing between ui and M2, and 2/1 and y 2 . u = Mi + JMa = y y = m/0 u , m 2 = Mi 2 + M2 2 . 1 = yi + jt/2 • Mi M2 u Mi 2 + M2 2 J M1 2 + Ma 2 Vi = Mi , 5 and y 2 = - M2 M 2 ' Mit/i + Mat/2 = Mi 2 — M2 2 _ - M2 2 2 + M2 2 M 2 = 1 + 2 M22/2 uiy 2 - M2V1 = - 0M1M2 . 2— 5- = +2mi2/ 2 . Thus and (9) REDUCTION IN FORM OF THE POWER EQUATION 201 Introducing these expressions into equation (8), P = P (cosh 2 al cos 2 pi + sinh 2 al sin 2 pV) + (eo 2 Vi + io 2 u{) cosh al sinh aZ — (eo 2 2/2 + *o 2 t*2) cos j8Z sin pi \Po -f 2 [P0W22/2 + (eoV - e</V) uiy*]\ (cosh 2 aZ sin 2 0Z + sinh 2 al cos 2 01) . (10) Combining the first term, and the portion of the fourth term in the right hand member which contains Po alone, there results P [cosh 2 al (cos 2 pi + sin 2 pi) + sinh 2 al (cos 2 pi + sin 2 pi)], which reduces to Po (cosh 2 al + sinh 2 al) or P (1 + 2 sinh 2 al). The expression *%" - e "io' = P i (11) represents the wattless volt-amperes at the load end (see Stein- metz' "Alternating Current Phenomena," page 218), and is commonly designated by the symbol, PK Numerically, also, Po ? '= eo^sin 0o, where <£o is the power-factor angle of the load, counted, in the vector diagram, from E to I in a counter-clockwise direction, and is thus positive in value for a leading (condensive) load. The real power is Po = eoio cos <f>, so that volt-amperes = e «o = Vp 2 + P > 2 . In further writing, products of the form UV COS (0„ — 0„) = U1V1 + W2V2 may be written in the abreviated notation, U • V; products of the form uv sin (0„ — 8 U ) = U1V2 — U2V1 may be written with the notation, U X V. U . V ■ UV COS (&x — 6v) = U\Vi -f W2«>2, £7 X 7 ■ uv sin (0„ — 0„) = U1V2 — U2V1, 202 POWER RELATIONS IN LONG LINES where U = Ui + jut, V = Vx + JUj, UV ss (u i+jifr) ( vi +jihh_ uv = Vui 2 + U3 2 Vy x 2 + v, 2 . Therefore (uvy = (U-V)*+(Ux V)\ For example, P = E'l, P' = E X /, and volt-amperes = ei. 123. Final Form of Power Equation. — For further reduction cosh al sinh al = \ sinh 2 aZ, cos/3Zsin/3Z = 5sin2j8Z, and cosh 2 al sin 2 /SZ + sinh 2 aZ cos 2 /3Z = sinh 2 aZ + sin 2 /3Z, so that the final form may be taken to be P = P (1 -f- 2 sinh 2 al) + (eo^i + *o 2 Wi) sinh al cosh aZ — (eo 2 ^ + *o 2 M2) sin /SZ cos # + 2 (P0U22/2 + Po'ttijfc) (sinh 2 «Z + sin 2 /SZ). (12) 124. Numerical Example. — As an illustration of this final equation, let us compute the power at- the generator for a 100 mile transmission line with the constants r = 0.275, g = 0.15xl0-«, L= 0.00204, and C= 0.0146 Xl(H per mile, when the e.m.f. and current at the load end are respec- tively 100,000 volts and 200 amperes, and the load power-factor angle is <£o = —25.0 degrees. This signifies a "lagging load." Let the frequency be 25 cycles per second, and in order to use the equation in its general form, let neither Eo nor Jo be selected as reference vector, but assume Eo - eo' + jeo" = 100,00 0/65.00 deg. = 42,262 + j 90,631 volts, r- U' + jV = 200.0 /40.00 deg. = 153.208 + j 128.558 amp. Then Co = 100,000 volts, to = 200.00 amperes, ^o = - 25.000 degrees, NUMERICAL EXAMPLE 203 and Po = eo'io' + e"W = eoio cos <£o = 18,126,300 watts, iV = e<>%" — eo"io' = e<#o sin <f> = — 8,452,400 volt-amps., tf-v/f- - Mi + JM2 = 406.61 — .; 135.63 ohms, jj = y ^ = ft +jy 2 = 0.0022131 + j 0.00073821 mhos. From Table XIX, al = 0.037205, &l = 5.2264 degrees. Substituting these numerical values in equation (12), the four separate terms which go to make up the total power at the genera- tor, P, are, in consecutive order: First term + 18, 176,700 watts Second term + 1,429,810 " Third term - 177,520 " Fourth term - 84,280 " P = 19,344,710 watts. Computing the generator e.m.f. and current by equation (19), Chapter VI, we have E = 41,892 +.7*98,894 volts, J = 132.10 + j 139.65 amperes, and the power P = eiii + e 2 * 2 *" e * cos = 19,344,500 watts, which verifies completely the value obtained from equation (12) If the line loss be roughly computed by P' = ioh-l + e<?gl, we obtain P' = 1,250,000 watts, while the true value of the line loss is P' m P - P = 1,218,410 watts, the last three figures of which are unreliable, since the computa- tions were made with five-place logarithm tables. 204 POWER RELATIONS IN LONG LINES 125. Line Loss. — For very accurate work, it may be desirable to use the formula P' = P-P = P 2sinh 2 aZ + (eo 2 2/i + io 2 Ui) sinh al cosh al — (eo 2 2/2 + to 2 M2) sin 01 cos 01 + 2 (P U2i/2 + Po y Uiy f ) (sinh 2 al + sin 2 0Z), (13) to calculate the line loss directly, for this obviates the numerical inaccuracy involved in taking the difference between two large and nearly equal quantities, in the case of a line whose losses are small in comparison with the power transmitted. With long telephone circuits, on the other hand, the loss in the line may be many times the power received at the load end, so that values of power loss for a given load may be conveniently obtained either by numerically subtracting, P — P , from equation (12), or by the use of (13) where the subtraction has been made in the analytical expression. 126. Unloaded Line. — For an unloaded line, P = 0, the expression for P is also the expression for line loss, and it is P = {&?V\ + fco 2 Wi) sinh al cosh al JPo=0 — (eo 2 2/2 + io 2 ih) sin 01 cos 01 + 2 Po'Wii/2 (sinh 2 al + sin 2 01) . (14) The equation as given may be applied to a line supplying a load of zero power-factor, so that P = 0. Po } is then numerically equal to eo^o — positive for a leading load and negative for a lagging load. If both P and P y are zero, then the line must be either short-circuited or open-circuited (e = or in = 0), and for these conditions, equation (12) becomes P = to 2 (wi sinh al cosh al — ih sin 01 cos 01) , Jeo=0 P\ — ^ (yi sinh al cosh al — t/ 2 sin 01 cos 01) . J«o=0 (15) 127. Power in Distortionless Lines. — In Chapter VII, the quantity V = a + j0 was discussed, and by equations (13) and (14) it is seen that for all frequencies, the attenuation constant, a, and the velocity of propagation, 8, are constant, and the wave- POWER IN DISTORTIONLESS LINES 205 length constant, /3, is directly proportional to the frequency when the relation gh = rC (for distortionless line) (16) is fulfilled. A line whose properties are thus related is said to be a "distortionless line." The reason for this name is quite obvious — the constancy of a and S for all frequencies. If an alternating wave is made up of a number of harmonic frequencies, all of these different frequencies in either an e.m.f. or a current wave will be propagated along such a line with the same attenuation and the same velocity. Hence the wave form of the propagated wave does not change, all of the component harmonics retaining the same relative phase position throughout the line, and all decreasing in the same proportion. This, of course, is true only for the com- ponent (main and reflected) waves, and not for the resultant wave, except in the case of an infinitely long line, where the main wave and resultant wave are identical. This matter of wave distortion is of the utmost importance in telephonic work, and will be dis- cussed in connection therewith. In such a distortionless line, then, by equation (16), g r where k is the proportionality constant. Then Z = r+jwL = r + jwkr, J . * Y = g+ju } C = g+ja>kg,\ KU) so that "-v/f-vMBI-v^ This equation, interpreted, shows that for a distortionless circuit, the infinite line impedance, U, contains no imaginary component, and has a numerical value equal at all frequencies to the infinite line resistance for direct current, y - • U = U1+JM2 = V"' JgL-rC ' g y] = yi +jy 2 =\fl, JgL=rC Y T uJ = 0, y 2 \ =0. JgL=rC JgL=rC (18) 206 POWER RELATIONS IN LONG LINES Introducing this condition, that both ut and y 2 are zero, into the general equation for power, (12), there results, for the distortionless line, J l - JgL-rC Po (1 + 2 sinh 2 aZ) f . + ( e ° 2 \~ r + *° 2 V ") sinh al cosh al > ( 19 > a very simple and interesting form. Since cosh 2 al = 1 + 2 sinh 2 al it is seen that equation (19), for a load power-factor of unity, is identical with equation (26) of Chapter II for a direct current line. In this case, as has been shown to be true for any frequency, -1 " so that, finally, p] - Po cosh 2 VgTl + ± (e 2 V Q - + io 2 y -) sinh 2 V^Z. (20) This resulting value of power at the generator end for a given power, Po, at the load end, is entirely independent of the frequency, in so far as the fundamental line constants r, L, g, and C are in- dependent of the frequency. Consider the numerical case used as illustration of equation (12) in the last paragraph. For this line, gL = 0.0003060 X 10"« and rC = 0.0040150 X 10 -6 . Suppose g, r, and C to remain constant, but by some means the value of L is increased until the relation, gL = rC, is fulfilled. The line then becomes distortion- less, and equation (20) applies. The self-inductance, L, must be increased from 0.002040 to 0.0267667 henry per mile to bring about this condition. Then, for the same load for which the preceding numerical calculation was made, we have for the power at the generator, by equation (20), P = 19,391,400 watts and P' = 1,265,200 watts = line loss. The line loss with the original value of self-inductance per mile was 1,218,400 watts, indicating that, in this case, the creating of LOAD IMPEDANCE FOR A MINIMUM LINE LOSS 207 a distortionless circuit by an increase of the self-inductance does not bring about a decrease in the line power loss. From this particular numerical example it is not possible to generalize, however, to the extent of saying that the creation of a distortion- less line by an increase in the self-inductance will always cause an increase in the power loss for a given load. It is possible to differentiate equation (13) with respect to either L or C (remember- ing that the auxiliary constants, Ui, Vq, y\, yi, a, and /3, are func- tions of r, g, L, and C), and thus to determine for what values of the independent variable the line loss would be a minimum, but the resulting expression after differentiating would be so com- plicated as to prevent any practical results from being obtained. For a given load, and either L or C variable, it would be best to plot a curve showing the power loss as a function of the variable, and from the curve to estimate the value for a minimum loss. 128. Load Impedance for a Minimum Line Loss with a Given Po. — In transmitting power over a leaky line by means of a direct current, a maximum efficiency of transmission occurs when the load-end resistance (equivalent resistance of the load) is equal to the infinite-line resistance y -, according to equation (34), Chapter II. With this value of load resistance, the efficiency of transmission is constant and independent of the magnitude of voltage and current. A similar condition obtains in the case of a line carrying alter- nating currents. As an introductory case, suppose an amount of power, P , is to be delivered at the load end of a line, at a power- factor, cos <f>o. This constant amount of power may be supplied by current at any voltage, the necessary relation being eoio cos 0o = -Po, or e io = Po COS 0o What must the values of e and i be in order that the line loss be a minimum? Let 2 = -r = scalar value of load-end impedance. to Then if = — ^— and e 2 = ^» (21) Zocos<£o COS 0o 208 POWER RELATIONS IN LONG LINES Since P and fa are assumed constant, P y = P tan </>o is also a constant. Thus, if W represent the variable portion of the line loss as eo and io are varied, with constant Po and <&, we have, from equation (13), W **\ (eo 2 yi + io 2 Mi) sinh 2 al - £ (eo 2 y 2 + t'o 2 ^) sin 2 /3Z, and by equation (21), W = ^(-y^-+— ^-)sinh2aZ 2 \cos0o Zo cos 0o/ _Po/_^. _J^.\ in ^ ( 2 \COS0o ZoC08(f>o/ v ' In order that this variable portion of the line loss, and thus the total line loss, be a minimum, differentiate equation (22) with respect to Zq and equate the derivative to zero. dW Po / Mi\ . , „ , ^ = T sec<^-- 2 )sinh2aZ - ^ sec <t> (y 2 - §1 sin 2 0Z = 0. (23) Solving (23) for z , 2 _ U\ sinh 2 aZ — tt2 sin 2 01 _ U\ sinh 2 aZ — iiy sin 2 ffl 2 ,_ .. 2/i sinh 2al — y 2 sin 2 $ Mi sinh 2 a£ + W2 sin 2 /3i The value of Zq given by equation (24) is independent of the amount of power delivered, as well as of the load power-factor. That is, for a given amount of power delivered, to secure a mini- mum line loss under any condition, the load impedance should have the above numerical value. For varying amounts of deliv- ered power and varying power-factors, such a condition demands, therefore, a variable and varying generator voltage — a condition which is usually not feasible, in power transmission at least. Notice that, in regard to the above statement of minimum line loss under any condition, an absolute minimum of line loss is not implied. For a given amount of power at a given power-factor the line loss is a minimum when «o is determined by the above equation, but obviously, by equation (13), for a given power Po and load impedance Zo, the line loss may be changed by varying the power-factor angle, <ft>, of the load. NUMERICAL ILLUSTRATION 209 129. Numerical Illustration. — As illustration of equation (24) consider the 400 mile, 60 cycle line, whose constants are given in Tables XII and XIV and some operating characteristics of which are given in Tables XV and XVI. A summary of the con- stants, and the calculations leading to a value of Zq by equation (24) is given below in Table XXVI. TABLE XXVI Determination of z for Minimum Line Loss, by Equation (24) r . 275 ohm per mile I 0. 15 X 10 -6 mho per mile 0.00204 henry per mile C . 0146 microfarad per mile I 400 miles f 60 cycles per second 2 at 0.312615 2 01 95.5384 degrees U = Ui +ju 2 385.143/-9 03' 26". 9 = 380.34 -j 60.630 ij = yi +jyt 0.00259644/9° 03' 26". 9 - 0.0025641 +j0. 00040875 sinh 2 al 0.31773 sin 2 01 0.99534 Mi sinh 2 al 120.84 u 2 sin 2 01 -60.35 j/i sinh 2 al 0.00081468 y 2 sin 2 01 0.00040684 Numerator 181.19 Denominator 0.00040784 zo 2 444260 Zo 666.53 ohms It thus appears that, for this line, a minimum line loss will be secured when, for any load, the generator voltage is so adjusted that the ratio, Zo, between the numerical values of load voltage and load current is 666.53. For example, to supply a load of, say, 4000 kilowatts at a power-factor of cos 25 degrees = 0.9063, the load voltage should, by (21), be eo = V *<> t = V 666.53 ' ' = 54,238 volts, ▼ cos <£o v 0.9063 and the corresponding load current is zo = — = 81.373 amperes. Zq 210 POWER RELATIONS IN LONG LINES Assuming this to be a lagging load, we have, with fa = —25.000 degrees and P = 4,000,000 watts, Po' = Po tan <fo = - 1,865,200 volt-amperes. By equation (13), P' = line loss = 951,030 watts. P = power at generator = 4,951,030 watts. p Efficiency of transmission = y = -£ = 0.8079, or 80.79 per cent. Note that this value of efficiency corresponds approximately with that for the 80 ampere entry in the first part of Table XV. 130. Line Efficiency. — As before pointed out, if the load impedance remain fixed, the current and voltage throughout the line change directly as the generator voltage. All powers and volt-amperes then vary as the square of the generator voltage, so that the efficiency remains constant. Since for any value of power, Po, a minimum line loss, and therefore a maximum efficiency of transmission, is obtained with the above value of Zo, it follows that each particular power-factor of load has a definite and character- istic maximum possible efficiency, viz., that corresponding to the value of Zo by equation (24). A curve may thus be plotted show- ing the maximum possible efficiencies as a function of the load power-factor, and such a curve is independent of the line voltage employed. p Let r] = efficiency of transmission = -^ and Z = Z p/— <fo . The angle of Z is taken negatively because an impedance with a positive angle, r + jx = z/^ , corresponds to a negative power- factor angle (lagging) load, and vice versa. In this discussion, <fo is the power-factor angle — the angle of 7 with respect to E . Then, by equations (12) and (21), — =1 + 2 sinh 2 al + 2 (uzy2 + Uiy 2 tan <fo) (sinh 2 al + sin 2 fil) + 2^b [(■* + 1) sinh 2 al ~ (*» + 1) sin 2 "J • (25) If </>o be kept constant in the above equation for efficiency, and the first derivative with respect to Zq be equated to zero in order to determine the value of Zo for either a maximum or a minimum, the result given by equation (24) will be obtained. Substituting LOAD POWER-FACTOR FOR MAXIMUM EFFICIENCY 211 (24) for Zq in equation (25), the efficiency expression for this special load impedance becomes: —1 =1 + 2 sinh 2 al + 2 Uiy 2 (sinh 2 al + sin 2 01) V Jzo by (24) + 2 U\\)i (sinh 2 al + sin 2 01) tan 0o — - Vuij/i sinh 2 2<xl + Utfji sin 2 2 01 (26) COS 0o 131. Load Power-Factor for Maximum Efficiency. — Differ- entiating with respect to 0o, 'ft) j = 2 Wi?/2 (sinh 2 aZ + sin 2 01) sec 2 0o o<Pq + sec 0o tan 0o Vui2/i sinh 2 2al + wyi sin 2 2 01. Equating the first derivative to zero, in order to determine 0o for a maximum or a minimum, and solving, 2M 1 y 2 (sinh 2 o!? + sin 2 j3Z) sin 0o = 7 ==> (27) vuiyi sinh 2 2 a J + W22/2 sin 2 2 01 for a condition of maximum efficiency, when taken in connection with the value of Zq by equation (24). To be rigorous, it is neces- sary to determine that the value of 0o by (27) corresponds to a minimum of — and thus to a maximum efficiency, rj. For such to be the case, the second derivative of — with respect to 0o v must be positive for the value of 0o given by equation (27). Differentiating again, *(t) , = 2 Uiy 2 (s'inh 2 al + sin 2 01) 2 sec 2 O tan O U0o + Vinyi sinh 2 2 al + 1*22/2 sin 2 2 01 (sec 3 O + tan 2 O sec 0o) = sec 3 0o { 4 M1I/2 (sinh 2 al + sin 2 /3Z) sin O + (1 + sin 2 0o) Vmyi sinh 2 2al + my* sin 2 2/8Z}. Substituting for sin 0o the value given by (27), (P (7) «6 by (24) <fr> by (27) d0o 2 [Mii/i sinh 2 2 al + v^yt sin 2 2 01 - 4 1*1 V (sinh 2 al + sin 2 #) 2 } sec 3 0o Vuiyi sinh 2 2 al + W22/2 sin 2 2 01 (28) 212 POWER RELATIONS IN LONG LINES It remains to be demonstrated that the value of the expression in equation (28) is always positive. From purely physical con- siderations, we know that if there are no line losses at all, the efficiency of transmission will always be unity, and therefore there will be no maximum or minimum and the second derivative as well as the first derivative have zero values. When there are line losses, the efficiency must reach a maximum value somewhere between its two zero values corresponding to leading and lagging loads of zero power-factor, but of the proper impedance Zq to make equation (28) applicable. Sin fa by equation (27) must then be real, from which we know that the quantity under the radical must be positive. The value of the radical itself is essentially positive, since the scalar value of Zq by (24) must be positive. Sec 3 fo is always positive, so the value of the second factor in (28) is always positive. The quantity in the brackets remains in question. To show that the quantity in brackets is always positive appears rather complicated, but some special cases covering the widest possible range of conditions possible in transmission lines may be investigated with ease. For a distortionless line, gL = rC, the value obviously is positive, since by (17) u^ and 1/2 both vanish, and Mi and y\ are positive. Again, suppose that g = and L = 0, a condition represent- ing maximum distortion. Then Z = r+j0 and Y = +j2*fC = + jb. vy V b c — =- V 26 v 1 » - * - v Tr 1 V = a + j/3 = Vjrb = Vrb/45° = ^ /t+ivf «-,-vf Then uiyi = \, ikyi = -\, and UiV = \. Substituting these values in equation (28) and denoting by B the NUMERICAL ILLUSTRATION 213 value of the expression in brackets, we have, when x represents the common value of al and /3Z, B = % (sinh 2 2 x - sin 2 2 x) - (sinh 2 x + sin 2 x) 2 . Expanding in a series, The coefficients of the series all being positive, the value of B is always positive, for this special case, and therefore the value of the second derivative of — is positive. V If we take the other extreme, that is, r = and C = 0, the same result will be obtained. Since for the above three conditions as widely separated as possible the value of <£o given by equation (27) corresponds to a maximum value of transmission efficiency as desired, it is reason- able to assume that the equation is correct under all conditions. Several numerical examples with intermediate conditions have shown this to be true. 132. Numerical Illustration. Variation of Efficiency with Power-Factor. — Applying equation (27) for O to the numerical case under discussion, see Table XXVI, the computation appears as below, in Table XXVII. TABLE XXVII Determination of <t>o foe Maximum Efficiency of Transmission Values from the logarithmic computation for Table XXVI log uiyi sinh 2 2 at 8 . 99322-10 * log u 2 y 2 sin 2 2 /3Z -8 . 39008-10 log sum of above 8.86864-10 log denominator 9.43432-10 log numerator 9.25072-10 * log sin 0o -9 . 81640-10 *o -40.938 degrees cos<fc> 0.75542 * The minus sign before the logarithm does not signify a negative logarithm, but that the num- ber represented thereby is negative. The angle of Zo is of opposite algebraic sign from that of <fo, so that Zo = Zo cos <£o — jzo sin #o. 214 POWER RELATIONS IN LONG LINES From Table XXVI, «o = 666.53 ohms, which gives Z = 503.51 +j 436.74 ohms, for a condition of maximum efficiency of trans- mission. It thus appears that, for this line, a maximum efficiency of transmission is obtained when the load power-factor angle is —40.938 degrees; that is, for a lagging load having a power-factor of 75.542 per cent. The numerical value of the load impedance being fixed by equation (24) and Table XXVI, the separate com- ponents thereof are determined as indicated in Table XXVII. n = 503.51 and x = 436.74 ohms. With this value of load end impedance, the efficiency always remains constant at its maximum value, regardless of the gener- ator voltage employed, and the power received varies, of course, as the square of the generator e.m.f. The efficiency for the above value of Zq is, by equation (25) or (26), 81.550 per cent. Numeri- cally, equation (26) is — = 1.020869 + 0.17812 tan <fo + 0.27184 sec 0o, or 1 9 1.020869 + 0.17812 tan O + 0.27184 sec *,' from which the following values were obtained: (29) TABLE XXVIII Transmission Efficiency of 400 Mile, 60 Cycle Transmission Line z = 666.53 ohms. Values from equations (26) and (29) <I>q = load power-factor angle. »? = line efficiency <t>0 V deg. -90 0.00000 -80 0.63444 -60 -40.938 0.79615 0.81550 Lagging -40 0.81547 -20 0.80301 0.77357 20 0.72728 40 60 0.65565 0.53388 Leading 80 0.27804 90 0.00000 NUMERICAL ILLUSTRATION 215 Fig. 67 shows the curves plotted from the above data. It is to be seen that throughout the possible ranges of power-factor, a much better efficiency of transmission is obtained with a lagging load than with a leading load. For the quite common power- factor angle of 40 degrees (P.F. = 0.766), the efficiency of trans- mission has a value with a lagging load of 81.55 per cent, while at this same power-factor and a leading load, the efficiency is only V 1.0 0.0 0.8 oT 0.0 0.5 0.1 0.3 0.2 Lbj Sin I L JlKl. 0.1 I .eai ing Lou d. I 'owe L*4* igle D «W es. -90-80 -60 1 -40 1 "t -0 1 f 40 1 T 80 & Fig. 67. — Transmission Efficiency with 2o Constant at its Maximum- efficiency Value. Curve Plotted from Equation (29). 65.56 per cent — a difference of 16 per cent. This being the case, it is obvious that from the standpoint of line losses, or efficiency, a lagging load is much to be preferred to a leading load, although in practice, the latter is viewed with favor on account of the superior voltage regulation obtained thereby. This is a direct result of the use of inherently constant voltage apparatus, and the conse- quent desirability of maintaining as nearly as may be a condition of constant voltage transmission. Theoretically, the ideal con- dition of transmission would be that for which the load power- factor would remain constant, while the voltage, as well as the current throughout the line, would vary as the square root of the power transmitted. From such a condition, a maximum economy in the use of power would result. 216 POWER RELATIONS IN LONG LINES For this particular line, then, the ideal conditions (as far as efficiency is concerned) may be summarized in a number of equa- tions, as follows: From equation (24) and Table XXVI, zo ■ 666.53 ohms. From equation (27) and Table XXVII, 0o - -40.938 degrees, and Z = 503.51 + j 436.74 ohms. From equation (26) and Table XXVIII, r] = efficiency of transmission = 0.81550. From equation (21), to = 0.044565 VPp and eo - 29.704 VW . At the generator, or sending end, p = f?= 1.2263 P . v From equation (19), Chapter VI, and Table XIV, i = 0.049348 VF and e = 32.894 Vp . i = \/J-i = i.i073to and e = J—eo = 1.1073 eo. From equation (19), Chapter VI, 4> = power-factor angle at the generator = +40.938 degrees. m z = - = apparent generator-end impedance = 666.53 ohms. I 133. Peculiarities Incident to Maximum-efficiency Trans- mission. — An inspection of these numerical values reveals the very interesting fact that, for a condition of maximum efficiency, the power-factor angle at the generator end is equal in magnitude, but of opposite algebraic sign from the power-factor angle at the load. The apparent impedance at the generator, z, is numerically equal to the load impedance, Zo. From load to generator, both current and voltage increase by the same percentage, and from the above, the ratio of similar quantities is seen to be '- - i - t/i- eo to '.% VARIATION OF MAXIMUM-EFFICIENCY CONDITIONS 217 The above statements have been taken from the numerical values for this particular line only, and though .the agreement of the numerical values through five significant figures is sufficient to establish their correctness beyond any reasonable doubt, general proof will be given later. (See section 138.) 134. Variation of Maximum-efficiency Conditions with Line Length. — The following table shows the numerical values pertaining to a condition of maximum efficiency of transmission over lines possessing the same fundamental constants per unit length as used in the above numerical illustration and of the various lengths tabulated. The constants are: / = 60 cycles, r = 0.275, g = 0.15 X 10^, L = 0.00204, and C = 0.0146 X 10 -6 . For convenience in determining the efficiency for load power- factors other than those corresponding to maximum efficiency, the constants entering into equation (26) are tabulated. They apply to equation (26) when it is written in the form, — = A + B tan <6 + C sec <*o. v TABLE XXIX Conditions Obtaining when Lines op the Previously Given Constants and of the tabulated lengths are operated at the Maximum Efpichcncy Possible Constants in Eq. 26 zo. ■ ohms 0Ot degrees Miles V A B C 1354.0 1.0000 1.00000 100 1243.9 -18.411 1.0008574 0.013788 0.04366 0.95944 200 1023.9 -31 . 175 1.003812 .052876 . 10214 .91641 300 821.3 -38.092 1.009946 .11082 .17964 .86857 400 666.5 -40.938 1.020869 . 17812 .27184 .81550 600 552.8 -41.054 1.03845 .24386 .37129 .75848 600 469.7 -39.184 1.06454 .29753 .47091 .69952 800 367.8 -31.066 1 . 14780 .33942 .65777 .58438 1000 328.4 -19.842 1.27570 .28583 .84210 .48361 1200 334.1 -10.052 1.44355 .18453 1.05731 .40247 1400 365.6 - 4.110 1.64193 .11800 1.30761 .33962 1600 399.9 - 5.431 1.86532 . 14971 1.58168 .29071 1800 416.5 - 8.616 2.1188 .28303 1.8892 .25083 2000 409.8 -11.790 2.4183 .45958 2.2493 .21645 Inf. 385.1 - 9.057 Inf. Inf. Inf. 218 POWER RELATIONS IN LONG LINES % 20 liog 1200 *0 1.0 1000 -40 x' -3B M 800 3 -28 O.r, 600 -20 O.J WO r> -16 -12 -8 o.e ■200 Liiu U- a*q i, M lies. —4 a * HJ 600 800 1 1000 1200 1400 1600 1800 20 Xf Fig. 68. — Values Obtaining Under Maximum-efficiency Conditions. A and c 2.6 i. B 0.52 2.4 48 2.2 0.44 2.0 0.40 1.8 / 0.88 1.6 «J 0*J2 1.4 hy> 'fc n-?fi 1.2 n?4 1.0 as 0.16 0.6 cm a4 0.08 02 0.04 Li le I eog thi i M lcs J 1 X) i 1 I * 800 1000 1 1200 1 1 1 14 JO vm 1800 a . Fig. 69. — The Constants in the Equation for Efficiency, 1 n A + B tan <to + C sec <fo. ZERO LENGTH LINES 219 Curves plotted from Table XXIX are shown in Figs. 68 and 69. In Fig. 68, for continuously increasing line length, the numerical value of z as shown for a condition of maximum effi- ciency decreases from a finite maximum at zero length to a finite value at infinite length. On the other hand, O , the load power- factor angle for maximum efficiency, increases rapidly in numerical value from zero for a fine of zero length to a maximum of about —41 degrees and then decreases, with a two-sided approach to a limiting value for infinite fine length. The maximum possible efficiency decreases continuously, approaching zero ultimately. 135. Zero Length Lines. — Special formulae are required for the determination of the tabulated numerical values for zero line length and infinite fine length, for in some cases the general formulae assume indeterminate forms. Consider equation (24), for the load-end impedance correspond- ing to maximum efficiency. For I = 0, this equation assumes the form .2 _ 2<) 2 P'=min. = J 1=0 To evaluate this expression, differentiate both the numerator and the denominator of equation (24) with respect to I for a new numerator and a new denominator, respectively. Thus, „ aui cosh — Buz cos yf ! _ , aui cosh -f- Biii cos or ZoV= mio .= uv/^^- (30) J i=o V aui + Bvq Further, by substituting a = V% (zy — xb-jr rg) and B = V% (zy + xb — rg), and also _ ag + Bb Bg - ab Ul ~ g* + 6 2 ' ^ " g* + 6 2 ' which follow from the relation _ t /Z VZY V a+jB • . we obtain as the reduced form of (30), 20 P'-min. = V -' (31) J 1=0 * g 220 POWER RELATIONS IN LONG LINES This remarkably simple expression for Zo applies with con- siderable approximation to lines of considerable length, as shown by the slow variation of this quantity in the first three entries of Table XXIX. The above value of Zq is numerically equal to the load-end resistance required for a maximum efficiency of transmission with direct current over a leaky line of any length, for which see equation (34), Chapter II. 136. Infinite Length Lines. — For a line of infinite length, equation (24) may be evaluated by dividing both the numerator and denominator by sinh 2 al. For infinite values of I, the fraction . , T vanishes, so that sinh 2 al Zo i»=min. = u. (32) J I-oo Therefore, with a very long line, to secure maximum efficiency, the load-end impedance should have a numerical value equal to u, the impedance of a fine of infinite length (see equation (13), Chapter VIII). For 1 = 0, equation (27) reduces to an indeterminate form. To evaluate, expand both numerator and denominator in series, re- taining only the first terms thereof. This yields ■M.1-— -~ "'^t + ^M =0. (33) Thus, for short lines, the maximum efficiency is obtained when the load power-factor angle is small, and for the limiting case of zero length, the angle should be zero. To evaluate (27) f or I = oo , substitute sinh 2 2 al = 4 sinh 2 al cosh 2 al, sin 2 2 01 = 4 sin 2 01 cos 2 01, and divide both numerator and denominator by sinh 2 al. 1+ _sin 2 Jtf_ sinh 2 al . sm<M = —uiy 2 —j= • (34) Ji,=max. / Uiyi coth 2 al + i^y 2 J,=max. f • sm 2 01 cos 2 [ft sinh 4 aZ The fractions in both numerator and denominator vanish for infinite values of I, and coth 2 al becomes unity. Therefore Sm <fx> „=max. = 7= = — = - — • (35) GENERAL 221 In this equation, — is equal to the sine of the angle, 0«, of the complex quantity U, which is the infinite-line impedance (see equation (13), Chapter VIII). Combining (32) and (35) we see that for very long (infinite) lines, the load-end impedance for maximum efficiency is Z ;=«, = U /~ d» = Mi — Ji;=max. M, (36) where U = y y = iti +juz = infinite line impedance. The imaginary component of Z in equation (36) is of the opposite algebraic sign from the imaginary component of U. This value of Z for maximum efficiency in a very long fine is not, as has often been assumed, that for which there is no reflected wave at the load end. From equations (10) and (10b), Chapter VI, the condi- tion necessary in order that the reflected wave be zero is that Z be equal to U, and this is not the same as the condition derived above. Therefore, for maximum efficiency, we can say that, in general, the load impedance must not be such that no reflected wave exists. 137. General. — It is easily seen from equation (27) that, for maximum efficiency, the load power-factor angle, <£o, is always either positive or negative for a line of given fundamental con- stants, and never varies in algebraic sign as the length of the circuit is changed. The algebraic sign of <£o is always opposite to that of 1/2, and is thus always the same as that of u^. The angle of the load impedance, Z , being equal and of opposite sign to that of <fo, we can say that, in a line of any length, the load impedance for maximum efficiency must always have an angle of opposite sign to the angle of the infinite line impedance, U. In power transmission lines, where the angle of Y is usually greater than the angle of Z, this means that the angle of the load impedance for maximum efficiency is usually positive (the angle of U being negative). Summarized, for maximum efficiency we have: t n If - < - , lagging (inductive) load. If — = — (distortionless line), non-inductive or unity power factor load. L C If ->-, leading (condensive) load. 222 POWER RELATIONS IN LONG LINES The above conditions depend only upon the line constants per unit length, and as before mentioned, are independent of the length of line involved. 138. Proof that for Maximum Efficiency the Ratio between e and e Equals the Ratio between i and io> and that the Genera- tor Power-factor Angle is Equal in Magnitude but of Opposite Sign to the Load Power-factor Angle. — In section 133, some general conclusions were drawn from a number of numerical relations observed in a particular case. These relations may readily be established as general propositions. As indicated by equation (16), Chapter II, the solution for a line in which distance is counted positively from the generator towards the load is obtained by substituting — I for I in the equa- tions based on a positive direction counted from load towards generator. Following through the developments leading to the power equation (12) for alternating current circuits, we find the substitution of — I for I to still be permissible. The substitution is also justified in equation (24), where we find the value of Zq un- affected by the change — both numerator and denominator chang- ing sign. This establishes the fact that, for a minimum line loss with a given power Pand power-factor angle <f> (at the generator), the scalar value of z is equal to the scalar value of Zo. Therefore, e eo e i - = Z = 2o = -r, Or — = — , i v e<> to for maximum efficiency. Substituting z for Zo, <£ for fo, - for t\ and — I for I in (25), V 77 = 1 + 2 sinh 2 al + 2 (1*22/2 + W1J/2 tan 4>) (sinh 2 al + sin 2 /3Z) -9-^if^ + -) sinh2aZ -(^ + -) sm2 ^i- & COS <p(\ z I \ z I ) 1 Comparing this equation with (25), we find the coefficient of cos <f> reversed in sign from the coefficient of — in equation (25). cos <po The coefficients of tan <f> and of tan <fo are alike in sign, so that, in equation (27), sin <f> will be of opposite algebraic sign to sin <fo. Such would not appear to be the case from an inspection of the equation for sin <fo alone, for the substitution of —I fori directly in (27) would not reverse the algebraic sign of sin fa. The double LOAD VOLTAGE FOR MAXIMUM ALL-DAY EFFICIENCY 223 sign should be placed before the radical in equation (27) if + and — values of I are to be used. In the reduction from equation (25) to equation (26), we have for the last term, involving cos <f> 0) by substituting the equation for Zo, St • u o 7 • o on t Ai sinn 2 aZ — 7^2 sin 2 /3Z < (y x sinh 2 al - y 2 sin 2 0Z) V . uo . ^ u ( ▼ Wi sinh 2 at + t<2 sin 2 /3t + + (wi sinh 2al — U2sin2 fit) v v V mi sinh 2 aZ — M2 sin 2 /3Z w The two radicals, which represent z , as well as u are essentially- positive, but in combining the expressions in parentheses to obtain the simplified form shown in (26), the algebraic sign of the entire term within the brackets is suppressed, if the positive sign only is retained before the resulting radical. Therefore, for maximum efficiency, <t> = — fo, and cos <f> = cos <j> . Then Since we have • ± 1 • ^ . 1 . ei cos <6 = - e<#o cos <6 , ei = - eoio. V V e eo z = - = Zq = —, i to to- The deductions drawn from the numerical results in section 132 are thus proved to be true in general. 139. Load Voltage for Maximum All-day Efficiency with Intermittent Load. — If a fine is to be operated so as to maintain constant load voltage and carries a given power load for a portion of the time and a negligible load for the remainder of the time (each day), the proper load-end voltage to ensure a minimum total daily loss in the line will be as determined in the following equations. In a given period of time, T, let t = the time during which full load is carried. Then T — t — time during which there is no load. Throughout the entire time, a constant e.m.f., e , is to be main- tained at the receiving end. By equation (13), the total energy loss in the line is W = t\P 2 sinh 2 al + § (e<?yi +i 2 Ui) sinh 2 al — h (eo 2 2/2 + itfvv) sin 2 /3Z + 2 (Pom* + PJuiyt) (sinh 2 al + sin 2 fit) ) + (T - (i «o 2 2/i sinh 2 al - \ efa sin 2 fl) . (37) 224 POWER RELATIONS IN LONG LINES The problem is, with a given power-factor angle fo, and power Po, what should be the value of e in order that W be a, minimum when taken over a fixed time 77 Placing ^ = k, and substituting, n i D 4. j. ' Po sec 00 /V = "o tan fa, to = — • eo Then ^r = A; P 2 sinh 2 al + ^ (eo 2 i/i + ^ Po 2 sec 2 <foui) sinh 2 al — 5 («o 2 2/2 + -3 Po 2 sec 2 ^0^2) sin 2 01 + 2 (P0U22/2 + Po tan 0oUi2/ 2 ) (sinh 2 al + sin 2 /3J) + (1 - k) % fa sinh 2al-y 2 sin 2 /SO eo 2 . (38) v . W Let £j = variable portion of -~-. The value of eo for a minimum of p is to be found. From equation (38), p = eo 2 (t/i sinh 2al — y 2 sin 2 /3Z) + \ (mi sinh 2 al - lit sin 2 00 fcP 2 sec 2 <fc>. (39) Co Differentiating with respect to eo, ■j- = 2 eo (2/1 sinh 2 a£ — 1/2 sin 2 #) deo o 1 (wi sinh 2 aZ - M2 sin 2 fit) kP 2 sec 2 <fc>. (40) eo Equating to zero to determine eo for a minimum p, '(ui sinh 2 a? — i*2 sin 2 01) kP 2 sec 2 <£o~l* «-[■ i/i sinh 2 ai — 2/2 sin 2 0Z (41) The above is the desired value of eo for which a minimum total energy loss will result. It is seen that the voltage necessary for such a condition varies as the fourth root of the fraction of time, k, during which the power P is to be used. Note that the product Po sec 0o is equal to the volt-amperes at the load. LINE LOSS WITH VARYING LOADS 225 Comparing (41) with (24), we see that if z = load impedance for maximum efficiency as given by (24), we may write e = (z 2 fcPo 2 sec 2 <£„)*. (42) Taking up the illustrative numerical example in section 129, suppose k = 0.60 — that is, the 4000 kilowatts of power is to be used only six-tenths of the time. By the above equation, then, e = 47,736 volts, as compared with 54,238 volts for maximum efficiency when the power is utilized the entire time. The above development is for a very special type of loading, but in the following a method for determining the proper load voltage for any load distribution will be given. 140. Line Loss with Varying Loads, and the Proper Load Voltage to Secure a Minimum of Such Loss. — Here, we again assume that the receiver, or load, voltage is to be maintained constant throughout all variations of load. The variable load must be given as a function of time, as by a load curve, for instance. We wish to determine what the constant receiver voltage should be in order that the total energy loss in the line, over the period of time in question, will be a minimum. If W represents the total energy loss during a given interval of time, by equation (27) and the transformations indicated in (38), during an interval of time, dt, dW = p 2 sinh 2 al + \ Uy, + ?£*?£** Ul ) sinh 2 al 1/ . , P 2 sec 2 tf> \ . * - g ( e ° # 2 "• ~2 **» J sin 2 & l + 2 (P M2?/2 + Po tan <^ Wi2/ 2 ) (sinh 2 al + sin 2 pi) \ dt. (43) Rearranging the terms and integrating between the limits and T, where T is the desired interval for which the total energy loss is to be a minimum, under the condition that eo remains constant, Podt o XT Pq tan 0o dt + \ (yi sinh 2al-y 2 sin 2 fit) e 2 T 1 C T + \ (mi sinh 2 al - v* sin 2 pi) —„ J P 2 sec 2 tf> dt. (44) 22(i POWER RELATIONS IN LONG LINES If, at all loads, eo is maintained constant, which is the condition for which equation (43) applies, we may differentiate this equation with respect to «o in order to determine the proper voltage for a minimum total line loss. dW —j— = (yi sinh 2 al — y 2 sin 2 pi) Teo 1 C T . (mi sinh 2 al - u* sin 2 pi) I P 2 sec 2 fo dt. (45) 6o Jo •A A *° 8 i 66 ■- 80 £ 2 75 X [Po» Jcfy 1 70 o 65 1.00 60 0.«»a \ 55 n.'.X) 11 _y s Co >♦, \ 50 0.85 10 1/ H ts a 0.80 9 / \\ 40 0.75 8 \ / rp„ \ 35 0.70 7 SO B 25 5 *> 4 15 3 ID 2 h 1 T me Ho urs t'roi n M idn Kht 1 { 8 l'O 1J2 1 '?"?"■! Fig. 70. — Illustrative Load Curve for 24 Hours. Equating the above value of the first derivative to zero in order to solve for the value of e which will give a minimum energy loss, (mi sinh 2 al - Uz sin 2 pi) / P 2 sec 2 <£o dt f ,. as eo = Jo I • (4o) (yi sinh 2 al - y 2 sin 2 pi) T ILLUSTRATIVE EXAMPLE FROM DAILY LOAD CURVE 227 By equation (24), Zq 2 may be substituted in the above for the fraction made up from the line constants, where Zq is the load-end impedance corresponding to maximum efficiency of transmission. Thus, eol = V7 (i f iV sec 2 <fc> dtf . (47) Po sec 0o = volt-amperes at the load, and since with a constant load voltage, the current is proportional to the number of volt- amperes, the value of the integral may be taken from charts produced either by a recording ammeter or a recording volt- ampere meter, after new curves, whose ordinates are equal to the squares of the ordinates of the original curves, are plotted. The integral between and T is equal to T times the average value of (volt-amperes) 2 . 141. Illustrative Example from Daily Load Curve. — As an illustration of this formula, consider again the 400 mile line which has previously been used. See Table XXVI. Let this be -used to supply a load whose value and power-factor for each hour from midnight to midnight are as given in the following table. The quantity, P 2 sec 2 <f>o = (volt-amperes) 2 , is calculated for each entry and listed in the table. Figure 70 shows the power, power-fac- tor, and (volt-amperes) 2 curves as plotted from Table XXX. The value of the integral may be derived by means of a planimeter, or, since the hourly values are ordinarily connected by straight lines in the load curve, an average of the 24 ordinates will yield the area divided by the base, which is the quantity desired. From Table XXX, 1 C T Average value of (P sec <ft)) 2 = ■=, J P 2 sec 2 <£<> dt = 62.8 X 10 12 (volt-amperes) 2 . From Table XXVII, using Zo for maximum efficiency = 666.53 ohms, in equation (47), eo = V666T53 (62.8 X 10 12 )* - 72,680 volts. This, then, is the proper value of constant load voltage to be used if the total daily energy loss in the line is to be a minimum for the assumed load curve. 228 POWER RELATIONS IN LONG LINES TABLE XXX Data for Illustrative Daily Load, Power-factor, and (Volt- amperes)* Curves Time counted in hours from midnight ( PoXio-* 008*o Po'sec'^oXlO-" 5.0 0.80 39.0 1 4.0 0.75 28.4 2 2.5 0.75 11.1 3 2.2 0.80 7.6 4 2.2 0.80 7.6 5 2.2 0.80 7.6 6 3.1 0.85 13.3 7 6.4 0.85 56.7 8 7.8 0.90 75.2 9 8.2 0.90 83.0 10 * 7.6 0.92 68.2 11 7.4 0.90 67.6 12 7.4 0.90 67.6 13 7.0 0.92 57.9 14 7.4 0.93 63.2 15 7.6 0.95 63.9 16 8.1 0.96 71.2 17 9.8 0.97 102.0 18 10.0 0.97 106.1 19 9.2 0.95 93.8 20 8.0 0.90 79.0 21 7.3 0.90 65.9 22 6.3 0.87 52.4 23 5.8 0.85 46.5 T = 24 1507.0 142. Dependence of Proper Voltage upon the Leakage Coefficient. — The preceding discussions have been illustrated by a line whose dielectric loss coefficient, or conductance, g, is 0.15 X 10 -6 mhos per mile. This corresponds to an energy loss of 500 watts per mile at a voltage to neutral of 57,700 volts, or a voltage between wires of a balanced three phase system of 100,000 volts. Since this leakage coefficient is the only line constant of comparatively uncertain value, it may be of interest to ascertain the proper load end impedances for maximum efficiency, as well as the proper load end e.m.f . to give maximum all-day efficiency for the above load curve, when different values of g are assumed. Table XXXI contains values of the more important quantities entering into the determination of Zo for maximum efficiency by equation (24) and of the proper voltage eo by equation (47) for the load specified in Table XXX, when different values of g are used. DETERMINATION OF RESISTANCE AND LEAKAGE LOSSES 229 TABLE XXXI Line Constants and Conditions for Maximum Efficiency in a 400 Mile Line with Different Values of g gXW 0.00 0.05 0.10 0.15 0.20 0.25 0.30 2al 0.28981 0.29741 0.30501 0.31262 0.32022 0.32784 0.33545 2# 95.752 95.682 95.608 95.538 95.468 95.402 95.336 Ui 379.54 379.83 380.10 380.34 380.56 380.75 380.92 «2 -65.819 -64.093 -62.365 -60.630 -58.897 -57.161 -55.420 z% 755.25 721.51 692.23 666.53 643.79 623.46 605.29 e 77,360 75,610 74,060 72,680 71,430 70,290 69,260 The above values are shown plotted in Fig. 71. 750 «0 X «| 2al io 8 710 0.35 78 720 77 .34 V 76 700 75 jn s" 74 680 €k 73 .:« W GOO 71 ja 70 010 % 69 .••so 68 0-20 \Zfl 67 .2!) 66 000 i 0.04 0.28 0.32 0.08 0.12 0.16 0.20 0.24 Dielectric Loss Coefficient, g, x 10 8 Fig. 71. — Conditions for Maximum All-day Efficiency for the Load Curve of Fig. 70, with Different Values of Leakage g. 143. Determination of the Resistance and Leakage Losses Separately; Graphical Method. — Equation (13) gives the total power lost in the line, which, of course, is made up of the sum of the i?r and e 2 g losses for every element of the line length. As it may sometimes be desirable to compute the losses due to each cause separately, formulae for this purpose will be given here. 230 POWER RELATIONS IN LONG LINES Two methods are available — graphical and analytical — the latter usually being preferable on account of the greater ease in application and the higher accuracy of the results. To determine the resistance loss for a given load by the graphical method, calculate the line current for a number of equidistant points along the line and plot, in polar coordinates, the curve representing i as a function of line length. The radius, R, in the polar curve is drawn to such a scale as to represent the line current. Suppose a unit angle in our diagram represents h units of line length — angles being counted in radians and distances along the line in whatever units we may select. Let a unit length of radius represent C amperes of current. Then, in the diagram, for any point in the line distant I from the load R = 1 i = CR and 6 = ^ , I = hB, (48) which are the relations between the coordinates of a point on the polar curve and the quantities in the line. The total resistance loss in the line is = r Ppdl, t/0 (49) where r is the resistance of the line per unit length (the same unit of length as used in defining h). The area of the polar curve is R? -i 2 dd. (50) If we substitute (48) in (49), the integration between limits zero and I is replaced by integration between limits and 0, and by this substitution, P r = rhC 2 f 6 R 2 M. f 6 R 2 dd. (51) By equation (50), however, the integral in the above equation represents twice the area of the curve, 2 A. Therefore, P r = resistance loss = 2 r^C 2 A. (52) In a similar manner, the total leakage loss may be determined by plotting a new polar curve representing the electromotive force at all points along the line. If C e is the number of volts repre- ILLUSTRATION OF GRAPHICAL METHOD 231 sented by a radius of unit length in the diagram, h is the number of units of line length per radian in the diagram, and A e is the area of the new polar curve, P, = 2ghCM., (53) where g is the conductance of the line per unit length. 144. Illustration of Graphical Method. — As an example, the power losses in the line whose voltage and current distribution are as given in Table XIII may be calculated. In the polar diagrams, 1 2 Fig. 72. — Graphical Determination of Separate Power Losses. or curves, let one degree of angle represent ten miles of line. Then h = !?? 10 = 572.96. Let one inch in the diagram of currents represent 40 amperes, and in the voltage diagram, let one inch represent 20,000 volts. Then C = 40 and C, = 20,000. For this line, r - 0.275 ohm per mile and g = 0.15 X 10 -6 mho per mile. Fig. 72 shows the current and voltage diagrams as plotted from the data in Table XIII. From these figures, by the aid of a planimeter, A = 15.13 square inches, and A e = 10.52 square inches. 232 POWER RELATIONS IN LONG LINES Therefore, P r = 2 X 0.275 X 572.96 X (40) 2 X 15.13 - 7,629,000 watts. P„ = 2 X 0.15 X 10-« X 572.96 X (20,000) 2 X 10.52 = 723,000 watts. P' = P r + P g = total line loss = 8,352,000 watts. The above value of P' added to P should equal the total power at the end of the 2000 miles. Forming the sum, we obtain 9,482,000 watts, which differs by 1.4 per cent from the value given in Table XIII. Considering that the quantities are all obtained graphically, the discrepancy is no more than might be expected. Nevertheless, a much closer result is obtained for the mid-point of the line. For this length, the areas of the current and e.m.f. curves are, respectively, 6.27 and 3.12 sq. in. Computing the separate losses and adding to P , we obtain 4,510,000 watts for the power at a distance of 1000 miles. The error here is less than one per cent. At best, the graphical process is difficult to apply on account of the necessity of determining the current and e.m.f. at a sufficient number of points along the line to enable smooth curves to be drawn. 145. Separate Losses Analytically. — To determine the re- sistance loss analytically, we must form the integral \ = r fi^dl, (54) Jo and this necessitates a knowledge of the square of the scalar value of current at any point in the line. At any point, I = ii+ ji*, (55) and in equation (5) we have given the values of t'i and U in terms of io r and V, the two components of the load current. Since the two components of I are at right angles to each other * = h 1 + k". (56) Equation (54) is then Pr==r X <* + *&* (57) SEPARATE LOSSES ANALYTICALLY 233 Substituting (5) in (57), after some transformation, P, ■ f I < t 2 cosh 2 al cos 2 (51 + i 2 sinh 2 al sin 2 j8Z «/o ( + e 2 -5 sinh 2 aZ cos 2 01 + e 2 -; cosh 2 al sin 2 /3Z it 2 w 2 + 2 io' (eoVi — «o"2/2) cosh al sinh aZ + 2 z'o" (eo'i/2 + e "t/i) cosh al sinh aZ — 2 V (e</2/2 + e "t/i) cos fil sin /3Z + 2 f " («b'yi - «o"?/2) cos /3Z sin /3Z > dl. (58) Introducing the relations, Po = e W + e "io" and P >' = eo%" - eo'V, and combining terms, P T = r f l j^o 2 + e 2 ^)sinh 2 aZ + i 2 cos 2 j3Z + e 2 -5 sin 2 flZ IF + (P yi + P '' 2/2) sinh 2 aZ - (Po^-Po'Vi) sin 2 0ZJdZ. 1 y Where -5 = yi 2 + 1/2 2 = scalar value of „• It- ^ Integrating, + 5- (PoVi + Po'y«) cosh 2 al (59) + -^ (P0I/2 - Po J 2/i) cos 2 0ZT- (60) 234 POWER RELATIONS IN LONG LINES Subtracting the value at the lower limit from the value at the upper limit, the final form becomes + ^(Po2/i + iV'i/2)sinh 2 aJ - 1 (Pw ~ Po'Vi) sin 2 fil I watts. (61) The equation for the total conductance loss may be written by symmetry from equation (61), with the exception that the algebraic signs before the terms containing P ' must be reversed. This arises from the fact that, in squaring the expressions for e x and e% in equation (4), products of the form to'eo" — «o"e</ replace the products of the form e</to" — eo'%' as obtained in the above de- velopment for P„ Thus, -£ e 2 dl = g \ - — (eo 2 + ioV) sinh 2 al (4a + -^(eo 2 -io 2 u 2 )sin2# + - (PqUi — Po'Uz) sinh 2 al a - | (P0M2 + Po'mi) sin 2 01 1 watts. (62) Equations (61) and (62) present the desired results. Though they give the losses separately, it must not be assumed that each of these losses is the same as the total line loss which would occur if the other were absent, for in the event of the conductance loss, say, being zero by virtue of a zero value of g, the value of P r by equation (61) would not be the same as for a condition of g being finite because of the change of a, /3, u, y h y 2 , etc., with the change in g. To determine P T when g = 0, all quantities entering the equation for P, must be determined for this value of g. This fact seems to have been overlooked by a number of writers who have developed formulae for power loss in lines. SEPARATE LOSSES AT NO LOAD 235 148. Illustration of Analytical Method. — As an illustration of these equations, let us calculate the separate losses for the line whose constants are given in Table XXVI, when this line supplies a load of 6000 kilowatts under maximum efficiency conditions. By Table XXVII, Zq = 666.53 ohms, and fa = -40.938 degrees, for maximum efficiency. e and t are then computed by equation (21), and Po 1 by the equation Po' = Po tan </>o. The main features of the computation appear below in Table XXXII. TABLE XXXII Calculation of Separate Line Losses Equations (61) and (62) Po 6,000,000 watts Po> -5,204,200 volt-amperes <j>o -40.938 degrees e 72,760 volts io 109 . 162 amperes For the four terms within the brackets Resistance loss Conductance loss No. 1 9,677,000 1,435,430 X 10 6 2 -2,838,190 421,000 X 10 6 3 835,660 123,960 X 10 6 4 -4,155,100 616,340 X 10 6 P r /r 3,519,370 P g /g 2,596,730 X 10 6 P r 967,870 P g 389,510 watts P = P r + Pg - 1,357,380 watts P = Po + P' = 7,357,380 watts 9 -Eff. = ^ = 0.81551 The above agrees with the value of maximum efficiency given in Table XXVIII as obtained from the efficiency equation (26). If we compute the resistance loss by the square of the load current times the total line resistance, 1,310,000 watts is the result. The square of the load voltage times the total line con- ductance gives 317,000 watts. The error in the approximation by this method is considerable. 147. Separate Losses at No Load. — Under no-load conditions, P = 0, and P ' = 0. Equations (61) and (62) then become very simple. 236 POWER RELATIONS IN LONG LINES Short Circuit sin 2 0ly p "I r / sinh 2 al sin2#\ . 1 = |/sjnh^_sJn2^A Open Circuit p ] r f sinh 2 al sjn2#\ 1 2 Ji =o 4\ a / (63) (64) Series expansions for the above formulae may be easily obtained, but no advantage would be gained — the above forms lending themselves admirably to the requirements of the computer. CHAPTER XI VOLTAGE REGULATION Case I. Both Components of Load Become Zero Simultaneously 148. Preliminary Developments. — Since most power genera- tion takes place under practically constant voltage conditions, and further, since the major portion of electrical apparatus requires essentially a constant voltage, regard must be taken of the fluctua- tions of the load-end e.m.f . of a line when the load varies in magni- tude. Adjustment as well as regulation of load-end voltage is often accomplished by means of over or under excited synchronous motors, which thereby consume large currents in quadrature with the impressed e.m.f. It becomes necessary, then, to be able to predetermine the magnitudes of such currents required to produce the desired effect. For any given receiver load, the voltage regulation of a trans- mission line is denned as, -r, . x . Rise in voltage from load to no load ,„ N Regulation = Load voltage (1) the generator e.m.f. being maintained constant throughout the change. At a given load J , and load-end voltage E , the generator voltage is E = E cosh VI + I U sinh VI, and if this generator e.m.f. be maintained constant while the load, and therefore 7 , is reduced to zero, the new value of load-end voltage becomes, by equation (8), Chapter IX, E ' = E sech VI = E + I U tanh VI. (2) This equation may be written, E = E ' - hU tanh VI, 237 238 VOLTAGE REGULATION a form which could have been written immediately from a gen- eralization discussed in a following section. The difference between the two load-end voltages is E ' - E Q = hU tanh VI. (3) 149. Scalar Values. — This difference divided by E does not give the regulation as defined in equation (1), for the definition is based upon the scalar values of the electromotive forces under the two conditions, while in (3) a vector difference is obtained. Using equation (3) in (1) would yield a complex expression, or value, for the regulation. The numerical value of this complex result would, in general, be greater than the true regulation, since the scalar value of the vector difference between two quantities is greater than the difference between the scalar values of the quantities themselves (unless the two quantities are in phase, when the results coincide). The expressions for line regulation must, then, be based upon scalar values. 150. Introduction of General Form of Equations. — The transmission line equations appear in the form, see equation (31), Chapter VI, E = AE + Bio, S - (fc + 3<h) E + (h + j&,) I , I-AI f + CE , I - (d + j(h) h + (ex + fa) Eo, A = cosh VI, B=U sinh VI, C = jj sinh VI. Under no load, E e 2 e 2 Eo' = ; — : — , «o' 2 = — r~ i o = -v (5) But E = AE + Bio, so that if Eo = e' + je", and I = i' + ji", , 2 _ {aie' - a^e" + bii' - H"Y + {axe" + o 2 e / + bd" + W ,„, eo - - 2 (pj By squaring, as indicated, and subtracting eo 2 = e' 2 ■+• e" 2 , eo'» - eo 2 = \ [bft' 2 + hH m + bH m + Wi' 2 a 1 + 2 (—diOze'e" + aibie'i' — aib&'i" — ajb\e"i' + a&e"i" - b x \hi'i" + a x a*Je" + aM'i" + a x b&"i' + aibxe'i" + (hhe'i' + bJhi'i")]. (7) (4) CASE I 239 Combining terms, and replacing by equivalents (see sections 122 and 123), eo' 2 - e 2 = \ \bW + 2 (<fa&i + 0262) P + 2 (0261 - afo) P Q 3 '\ . (8) Referring again to the form of notation described in Chapter X, section 122, where products of the form U1V1 -f- 1^2 are written U • V, and products of the form U1V2 — utf) \ are written U X V, a simplification in the equations may be made. Thus, equation (8) becomes eo'2 -eo 2 = \ \b*i<> 2 + 2A-BP -2Ax BP '\. (8a) - 151. Reactive Volt-Amperes Required at Load for Zero Regu- lation. — The above equation gives the difference between the square of the scalar value of the no-load voltage, eo' 2 , and the square of the scalar value of the load voltage, e 2 . If it is desired to so operate the system that the regulation as defined in equation (1) be zero, the amount of reactive power, or volt-amperes, Pq\ required to attain such a condition may be determined from equa- tion (8) by placing e ' 2 — eo 2 = 0. The voltage at no load will then coincide with the load voltage, numerically. i Q 2 may be eliminated from the equation by the relation eoto = Vp * + P >\ ; 2 = P ° 2 t 2 P ^ ' (9) Making this substitution, and solving the resulting quadratic in /V, fV = rj^ }ai&2— <kh ± V (ai&2-a2&i) 2 - ^[^o 2 +2 (axk+a^Pol j (10) or in simplified form PJ = |? j AxB±\f (AxB) 2 - ^ 2 |"^ Po 2 +2A. J BPo]| . (10a) 240 VOLTAGE REGULATION In the above equation, d = cosh al cos /SZ, Oz = sinh al sin 01, 61 = Ui sinh al cos fil — v* cosh aZ sin /3Z, 6j = Ui cosh aZ sin /3Z + Ut sinh aZ cos /3Z, 6» = 61 2 + W. (ID From these values, the combinations which enter in the equation for P ' may be written. Thus, A • B = aibi + a«&2 = U\ cosh aZ sinh al — Ui cos /3Z sin /3Z, i X B = 0162 — a 2 6i = ii2 cosh aZ sinh al + Mi cos /3Z sin /3Z, (12) 6 2 = m 2 (sinh 2 aZ + sin 2 01). In computing, values of the combinations only as given by the above formulae need be determined, and the labor involved is no greater than that required to determine a h a-t, 61, 62, and b 2 sepa- rately. As may, at times, be found desirable, equation (10) can be expressed in terms of the components of current in phase with and in quadrature with the load voltage eo. Thus, io p = in phase, or power, component = — , P } in' = quadrature component = — -• £o (13) As seen from an inspection of equation (10), two values of P ' or of io 7 are obtained for which zero regulation is secured. For power loads greater than a certain definite amount, depending upon the line properties and the load voltage eo, the quantity within the radical becomes negative, and therefore P J and itf become imaginary, which means physically that, under such con- ditions of line and load, it is impossible to secure zero voltage regulation by adding quadrature components of current to the load. 152. Conditions for Minimum Voltage Regulation. — Al- though under such cases as above it is impossible to secure zero regulation, the quadrature component of the load current may be so adjusted as to secure a minimum voltage regulation. The substitution of (9) in (8) gives * '2 _ eo *=I ^ + 2A.BP + V^-2A xBPod , eo 2 e z ) (14) CASE I 241 from which the change in voltage may readily be computed. To make this change, and therefore the voltage regulation, a minimum, differentiate with respect to Po 1 and solve for P ' after equating the first derivative to zero. d (e</ 2 - e 2 ) dP > Solving the above, after equating to zero, -*[*%* -* A * B ] (15) W2 sinh 2 al + Ui sin 2 01 , e P i] = AXB 2 = Jreg.=min. & 2 " 2 u? (sinh 2 al + sin 2 01) ]_ AxB _ Uz sinh 2al-{-Ui sin 2 ft reg.=min. " "b^ eo == 2 u 2 (sinh 2 ai + sin 2 ftZ) (16) For power loads below the critical value above which zero regula- tion cannot be secured, the value of Po 1 ' determined by the above formula will give a negative regulation — that is, the load voltage will be greater than the no-load voltage. 153. Conditions for any Voltage Regulation. — If we so desire, the quadrature component of load current may be so adjusted as to give any arbitrarily selected value to the voltage regulation. Let it be required to determine Po 1 ' to give a specified regulation, R. Then, if e = load voltage and e</ = no-load voltage, R = f°l^i°, eo ' m (1 + B) *. (17) Co The left hand member of (14) becomes eo '2-e 2 = (2R + R*)e 2 . (18) Introducing this value, and solving the quadratic in P } , ± y (Ax£) 2 -|^ 2 Po 2 +2A .£P -a 2 (2fl+# 2 )eo 2 l j- ( 19 ) This equation, like (10), may give imaginary values of Po 1 . With given line properties and load voltage e , real values of Po 1 will be secured through a greater range in P as independent variable when finite positive values of R are substituted in (19) than could 242 VOLTAGE REGULATION be obtained from (10). This should be obvious — it really amounts to saying that a larger amount of power can be trans- mitted with poor voltage regulation than with good. Again, two values of P ', for which the regulation has its specified value, are obtained from the equation. 164. Solutions in Terms of Load Admittance. — If the load admittance Y is given by Y = g + jb , then P = e o 2 0o, and P y = e 2 b . (20) It is often desirable to set up a solution in terms of 6 and g rather than in terms of e , Po, and P ' for then the solution is independent of the actual voltage employed and of the power transmitted. Substituting (20) in (19), b =^\AxB±V(AxBy-b 2 [b 2 g 2 +2A-Bg -a i (2R+R*)]}-(21) Placing R = 0, we get the special case given by equation (10), &ol = \ 2 { A X B ± V(A X B) 2 - &V - 2 b*A • Bg \. (22) The above values of g and 6 obtain under load, and must then become zero simultaneously, according to the bases for the develop- ment of the above equations. With constant generator voltage and a variable load Po, a variation of P J according to equation (10) will maintain constant load voltage — Po' disappearing simultaneously with P , but not being proportional thereto for intermediate loads. 155. Numerical Illustration. — As illustration, consider again the 400 mile, 60 cycle, power line described in Table XXVI. Using the line constants given in the various foregoing tables, equation (21) becomes numerically, 10 3 6 = 2.1141±V4.4693-10 6 ^o 2 -2132.4sr +5.6057(2 J R+« 2 ).(23) From this equation the values of 6 given in the following table were obtained. Both values of 6 corresponding to particular values of R and gfo are tabulated. Positive 6 indicates a con- densive (leading) load, and negative b an inductive (lagging) load component. CASE I 243 TABLE XXXIII Values of b = Load Susceptance Required to give a Regulation R with a Load Conductance go, when Both g and b Become Zero at No Load. R Defined by Equation (1), 6 X 10 3 Tabulated 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 4.228 4.115 3.973 3.796 3.571 3.270 2.800 Imag. Imag. Imag. 0.113 0.255 0.432 0.657 0.958 1.428 Imag. Imag. Imag. 0.04 4.334 4.226 4.092 3.927 3.721 3.454 3.077 Imag. Imag. Imag. -0.106 0.002 0.136 0.301 0.508 0.775 1.151 Imag. Imag. Imag. 0.08 4.438 4.336 4.209 4.054 3.862 3.621 3.299 2.790 Imag. Imag. -0.210 -0.108 0.019 0.174 0.366 0.608 0.930 1.438 Imag. Imag. 0.12 4.542 4.444 4.324 4.177 3.998 3.776 3.491 3.089 Imag. Imag. -0.314 -0.216 -0.096 0.051 0.230 0.452 0.737 1.139 Imag. Imag. 0.16 4.645 4.551 4.436 4.298 4.129 3.924 3.666 3.323 2.774 Imag. -0.417 -0.323 -0.208 -0.069 0.099 0.305 0.562 0.905 1.455 Imag. 0.20 4.748 4.658 4.548 4.416 4.256 4.064 3.838 3.525 3.096 Imag. -0.520 -0.430 -0.320 ^-0.187 -0.028 0.164 0.400 0.703 1.132 Imag. In Fig. 73 the values of b given in the foregoing table are plotted as functions of g . Multiplication of go and b by e 2 gives P and Po', respectively. For example, if e = 80,000 volts, and a load of 7680 kilowatts is to be supplied, g = 0.0012, and for a regulation of 8 per cent = 0.08, from the tabular data or curves, 6 = 0.003299 or 0.000930. Jtf then is 21,130 or 5950 kilovolt-amperes. The smaller is the only practical value to use, so the final load power- factor is cos <t>o = P.F. = cos tan -1 5950 7680 = 0.79, leading. 166. Nature of Curves Relating go and b . — The curves in Fig. 73 are ellipses, with their common center displaced to the left of the Y axis, and above the X axis by an amount — — r^ * A X B — — jn — . The standard form for the equation of an ellipse may be derived easily from equation (21) by placing A X B in the left hand member and then squaring. Proceeding in this way, we obtain as the final form, 244 VOLTAGE REGULATION The curves in Fig. 73 plotted from equation (24) are ellipses with unequal axes only because the scales used in plotting g and b are different. For equal scales, the curves become concentric circles, for (24) is the equation of a circle whose radius is Radius = £ (1 + R) - scalar value of cosh VI scalar value of U sinh VI (1 + 22). (25) 6, 10 4.8 4.4 4.0 3.B 3.2 2.8 2.4 Rer. ^20 hi. for iBi inn in 1 tl'LT. =0( o-ui 41 ] 1 \ J. u 1.0 u 0.8 I 0.4 0.8 1.8 Co* 10' a 2 4 (3 8 1 1 2 1 1 1JS Fig. 73. — Load Susceptance b Required to Give a Regulation R The coordinates of the center of the system of circles, with refer- ence to the origin for g and 6 , are A • B _ aibi + Q2&2 _ 1 Ui sinh 2 al — ity sin 2 /3Z 6 2 ^ = ~2 m 2 (sinh 2 oi + sin 2 01) ' and Ax5 aJh — 0261 1 W2 sinh 2al-\- Ui sin 2 /SZ b 2 2 u 2 (sinh 2 «Z + sin 2 #) (26) CASE I 245 157. Geometrical Construction of Curves. — From the in- formation in equations (25) and (26), the entire system of curves may be drawn very quickly with the aid of a compass. First, locate the common center of the circles with respect to the origin of coordinates, and then draw in the circles with radii determined by equation (25), according to the values of regulation, R, desired. For any given load, g and b , the regulation may be ascertained from the chart by measuring the distance from the center of the system of circles to the point determined by g and 6 . This length divided by j- gives (1 + R), and thus R. For the par- ticular problem in hand, the coordinates of the center of the circles are, by (26), • -0.0010661 and +0.0021141. By (25), for the radii of the circles we have Radius = 0.0023677 (1 + R) = distance from center of circles to origin +0.0023677 R. Figure 74 shows the family of circles drawn according to the above data. Reading from the center outwards, the curves are for 0, 0.04, 0.08, 0.12, 0.16, 0.20, and 0.24, values of regulation R. The curves are not numbered, for to do so would crowd the dia- gram unduly. The data from this set of curves should duplicate that given in Table XXXIII and plotted in Fig. 73. The power-factor angle of the load may be found graphically from the chart by taking the angle between the g axis and the line joining the point (g , b ) with the origin. Obviously, the scale of the diagram may be so changed as to read in amperes, or in kilowatts and kilovolt-amperes, for any particular value of e Q . Attention may here be called to the fact that this method of curve construction does not apply to transmission line problems only, but to any system for which linear equations of the form E = AE + Bio apply. Since the general electric circuit is represented by such an equation, the scheme applies thereto. Analytically, then, electric circuits are different only in the manner in which the coefficients A and B are different, in the above type form of equation. As before pointed out, the solution for an electrical 240 VOLTAGE REGULATION system amounts to the determination of these constants. It is to be noted in equation (25) that the coefficient of (1 + R) is the reciprocal of the scalar value of the short-circuit impedance. *,* w 4.8 1.0 3.2 ! i j 1 2.4 111 _ tor i l.fi // (1.8 i) % V 10 ■1 -1 fi ■4 a •1 v& 1 .0 2 4 ■■>. 2 0.8 -1.0 Item Bag fron CCIlt cr m 1'', n tWM • '. and 21 pi r M t ri't. ulut cm. Fig. 74. — Graphical Method for Plotting the Curves Relating ffo and 6 for Any Value of i2. 168. Relations between Vector Diagrams and Foregoing Curves. — The vector diagram from which data may be taken for the plotting of such curves as Fig. 73 or Fig. 74 is of interest, since it brings out in a clear manner the significance of the graphical construction just described. Fig. 75 is this vector diagram. If eo = load voltage, e ' = (1 + R) «o *= receiver voltage at no load, with regulation R. Since the scalar value of e (at the generator) remains constant, and e = a€o' = a (1 + R) e , we see that the scalar value of e to produce a regulation R must be given by the above equation. That is, under load, the end of the CASE I 247 vector E must lie on a circle drawn about the origin with a radius a (1 + R) eo. In Fig. 75, the vector AE is shown, with the family of circles for different values of R. To the vector AE are added values of BgoE , for various values of g , thus completing the vectorial representation of the equation E = AE + BI , for values of I = (goE ) in phase with the load e.m.f. This vector addition gives the points spaced along the line AEqM, correspond- A =0.69020 19I69S. BsJ89L61,l72lM, SeeiableuXIV Fig. 75. — Vector Diagram for Regulation Calculations. See Fig. 74. ing to different values of g . A wattless component of load current jboEo requires the addition of another vector jBb E to the vector from the origin to the point on the line AE M, located as de scrib ed. jBboEo is at right angles to BgoE , and the refore to the line AEqM. Drawing a line through the point on AEoM which corresponds to some particular value of g , we determine the numerical value of the quantity BboEo required to give a regulation R by the per- pendicular distance from the line AE M to the ntersections of the new line with the proper circle. These perpendicular distances are the intercepts on the new line, NQ, between the line AEqM and the intersections of NQ with the circles. These intercepts divided by beo give the required values of bo. In the diagram, a number of lines parallel to NQ are drawn, each one corresponding 248 VOLTAGE REGULATION to a particular value of g . By drawing the diagram to be Q times the voltage scale, distances along the line AEqM represent g directly, and distances along the line NQ represent 6 . The mechanical construction of the curves in Fig. 74 is accomplished by this method. If the above mentioned scale is used, distances along AEqM correspond to distances along the g axis in Fig. 74. The point AE Q in Fig. 75 is the origin in Fig. 74, and distances along the lines NQ correspond to distances along the & axis. The 6o axis itself is r epre sented by the line parallel to NQ drawn through the point AE . The origin in Fig. 75 corresponds to the center of the circles in Fig. 74. If coordinate paper is used in drawing the curves, as in Fig. 74, the desired data is immediately obtained, without the necessity of actually constructing the vector diagram, Fig. 75. 159. General. — In the above developments it has been assumed that the real power P and the wattless volt-amperes Po' both become zero at no load, in determining the rise in voltage at the receiving end from load to no-load conditions. When such is the case, P</ may, as has been shown, be so adjusted as to give zero voltage regulation, according to the definition in (1). If the reactive volt-amperes, Po', are consumed by a separate machine, or machines, it may happen that P and P J would not both become zero, but only the real power. Under such conditions, a different value of Po* is required to give a specified voltage regulation. If we assume that P ' remains constant while the real power, P , becomes zero, a value of P J can be determined for which the regulation will have a certain specified value. Actually, P ' is made up of two parts, one of which arises from the reactive volt- amperes provided to the load proper, and which, therefore, be- comes zero simultaneously with P , and the other which consists of the wattless volt-amperes supplied to whatever voltage regulat- ing device (as a synchronous condenser) is permanently connected to the load end of the line, and which, therefore, does not become zero simultaneously with P . If we consider the power required to drive the synchronous condenser, the situation becomes still more complicated. When such a condenser is used to consume a wattless component of current, a condition of constant reactive volt-amperes consumed thereby is not secured, for on removing the load, the voltage rises, and on account of this increased voltage, the synchronous condenser will consume a smaller wattless com- CASE II 249 ponent of current, if leading, and a larger wattless component, if lagging. The portion of Pn 1 ' supplied to the synchronous condenser is not constant with varying voltage; neither is the wattless com- ponent of current consumed thereby, nor the equivalent suscept- ance representing the same. To express, analytically, the per- formance characteristics of such synchronous condensers, and to incorporate these expressions in solutions for line regulation, is scarcely practicable, but in the following, a number of solutions, based upon different assumed load-end conditions, are given. Case I, for P and P ' disappearing simultaneously, has. just been given. Case II. Total Load-end Susceptance Constant 160. Load-end Voltages in Terms of Load-end Admittances. — Here we assume that the total load-end susceptance, 6 , remains constant while the load-end conductance, g , becomes zero at no load. Let Y = go + i&o = admittance under load. 1 Yq'= -\- jb = admittance at no load. J By equation (20), Chapter II, in general, for generator voltage E, E ° - coshVl + WsinhVl ~ l0ad v0ltage > (28) and since cosh VI = a\ + j(h and U sinh VI = 6i + j'62, we have, under load, E En = '° «i + j(h + (go + jb ) (61 + jb 2 ) and at no load (g = 0), Eh = di + j<h + jb (61 + jb 2 ) (29) Let R = voltage regulation, with constant generator voltage, e. Then e ' = (1 + R) e , and by substituting (29), e 2 (l + RYe* 60 — (a t - W>2) 2 + (02 + W>i) 2 (ai + 0o&i - 6 6 2 ) 2 + (02 + g &2 + W>i) 2 = (l + fl) 2 eo 2 . (30) 250 VOLTAGE REGULATION Clearing of fractions, combining terms, and substituting: fll 2 + 0^ = 0* and bf + k 2 = 6 2 , we obtain - 6* (2 R + «*) &o 2 + 2 (2 R + # 2 ) (o,6, - aA) &o + 2 (0,6! + 0262) f/o + 6V - a 2 (2/2 + fl») = 0, or, when written in simplified notation, - 6 1 (2 # + A 2 ) 60 2 + 2 (2 # + R?) A X B60 + 2A.^ + &W-a 2 (2i2 + /2 2 ) =0. (31) 161. Constant Susceptance Required for a Given Regulation. — For convenience in writing, let 2 R + # 2 = m, m + 1 = (1 + #) 2 . (32) Then, by solving (31) for b , and combining terms, 60 = ^ I A XB±\J(A x5) 2 -a 2 6 2 + 2^^ + ^o 2 |. (33) In equation (33) the constant term within the radical is (A xB)*-a?b 2 = -(A -B) 2 and therefore, equation (33) reduces to bo = ^AxB±>J±(b>g + A-By-(l+^)(A.B)^ (34) 162. Real and Imaginary Solutions. — In order that 6 be real, the quantity within the radical must be positive, i(6Vo + A. J B) 2 = (l + ^)(A. J B) 2 , (35) {Vg a + A-By + m — (A • BY — ' (36) from which, by (32), (l + fl) 2 s(l + J^|) 2 - (37) CASE II 251 Thus, for a real 6 with a positive R, b 2 and for a real 6 with a negative R, b 2 == # s j^g go, (a) -2 = R=-2-j^g . (b) (38) No real value of b can be found for a value of R which does not he within the above limits. In other words, for a given value of <7o, values of R not included within the above limits cannot be produced by shunting g with a constant susceptance, b . It is obvious from equation (33) that, for a finite g and R = 0, b becomes infinite. Under no condition, then, can zero voltage regulation be secured by such means, though as low a regulation as desired may be secured by the use of a condenser (or inductance, depending on the algebraic sign of bo taken from curves similar to Fig. 76) of sufficient size. These facts are emphasized, because in a recent engineering publication a method, based on incorrect ideas, was given for the determination of the constant value of JV to give zero regulation. Constant P } with zero voltage regu- lation implies a constant b , and according to the above, such a solution is impossible. 163. Numerical Illustration. — Before further development from equation (34), let us see the form of curves resulting there- from, when applied to the numerical example just considered. Table XXXIV, below, gives values of 6 computed from (34), which, for the line constants used, becomes KF&o = 2.1141± y i(10 3 sr + 1.06610) 2 - 1.13657(l +~). (39) Both values of b resulting therefrom are tabulated. •_\-)2 VOLTAGE REGULATION TABLE XXXIV Values of b = Load Susceptance Required to give a Regulation R with a Load Conductance, g , when g = at No Load and 6 Remains Con- • STANT. R as Defined by Equation (1). b X 10 s Tabulated 0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 0.0020 0* Inf. Inf. Inf. Inf. Inf. Inf. Inf. Inf. Inf. Inf. 0.04 -0.0259 4.2541 -1.2439 5.4721 -2.2394 6.4676 -3.1405 7.3687 -3.9890 8.2172 -4.8044 9.0326 -5.5972 9.8254 -6.3737 10.6019 -7.1379 11.3661 -7.8936 12.1218 0.08 0.8232 3.4050 -0.0602 4.2884 -0.8380 5.0662 -1.4860 5.7142 -2.0915 6.3197 -2.6707 6.8989 -3.2320 7.4602 -3.7809 8.0091 -4.3202 8.5484 -4.8522 9.0804 0.12 1.2793 2.9489 0.4274 3.8008 -0.1897 4.4179 -0.7292 4.9574 -1.2288 5.4570 -1.7045 5.9327 -2.1639 6.3921 -2.6128 6.8410 -3.0510 7.2792 -3.4849 7.7131 0.-16 1.6525 2.5757 0.7746 3.4536 0.2149 4.0133 -0.2631 4.4913 -0.7013 4.9295 -1.1160 5.3442 -1.5152 5.7434 -1.9036 6.1318 -2.2841 6 5123 -2.6584 6.8866 0.20 Imag. 1.0346 3.1936 0.5050 3.7232 0.0660 4.1622 -0.3317 4.5599 -0.7057 4.9339 -1 0643 5.2925 -1.4122 5.6404 -1.7525 5.9807 -2.0867 6.3149 0.24 I mag. 1.2495 2.9787 0.7311 3.4971 0.3178 3.9104 -0.0514 4.2796 -0.3962 4.6244 -0.7252 4.9534 -1.0436 5.2718 -1.3541 5.5823 -1.6590 5.8872 0.28 Imag. 1.4433 2.7849 0.9177 3.3105 0.5210 3.7072 0.1725 4.0557 -0.1502 4.3784 -0.4567 4.6849 -0.7524 4.9806 -1.0403 5.2685 -1.3224 5.5506 164. Nature of Curves. — Plotted, this data gives the family of curves shown in Fig. 76. The curves are hyperbolas which become narrower as R increases. The true shape of the curves is not apparent, since the g and b scales are necessarily taken un- equal in order to make the curves readable. From equation (39), it is obvious that the displacement of the axis of the hyperbolas from the g axis is 2.114 X 10~ 3 . While in the preceding case (g and 6 disappearing simultaneously) the value b = 2.1141 X 10 -3 gives a minimum regulation as pointed out, in this case, where & does not vanish at no load, an inspection of the curves shows that 6 = 2.1141 X 10 -3 is the value which gives a maximum voltage regulation, with a given value of g . The left-hand branches of these hyperbolas are not shown, since a negative value of g is meaningless. Rationalizing equation (34), (0o + ^) ! -(2ft + R*)(6o-^J=(l + fi)'(^) ! - (40) CASE II 253 This is the equation of an hyperbola whose center is at the point A>B + AxB (41) 6 2 '62 and whose axis is the horizontal line through this point (g being Fig. 76. — Relation between g and 6 for Any Value of R when 6 does not Vanish but Remains Constant at No Load. Equation (39). plotted horizontally). The vertices of the different curves then lie on this line, and their g coordinates are R A-B (42) for the different values of R. This value may be obtained from equation (38) by using the equality sign, which thus gives the minimum value of g for which a regulation R may be obtained. 254 VOLTAGE REGULATION The asymptotes to the curves pass through the center (equation 41), with a slope , =, and their equation is, therefore, v 2 R -{• R? . Ax B 1 / ,A-B\ .... The substitution of negative values of R according to equation (38) in equation (39) will yield the same family of curves as shown in Fig. 76. The curve for R = —2.16, for example, is the same as that for R = +0.16, but such solutions are meaningless for a negative regulation numerically greater than 1 has no significance. 165. Significance of the Coordinates of the Central Point. — In this problem, the center for the system of hyperbolas coincides with the center for the system of circles in the foregoing case, and attention is again called to its location in terms of the short-circuit impedance of the line. The general equation on which these dis- cussions are based is E = AE + BI , and if we make E — 0, we have h A { ** } By E is meant the resulting terminal e.m.f. at the load end, so that if we consider an e.m.f. applied at the load end while the generator end is short circuited (E = 0), we have B _ applied voltage at load end A current produced at load end = short-circuit impedance measured at load end. (45) The reciprocal of this quantity is A _ di -f- j(h _ a>ibi + (hbi _ . aJh — <hbi _A'B .A X B B~ &i+j&2~ P J V~ ~P~~ 3 ~tf~ Iy = \-y coth VI = short-circuit admittance. (46) 166. Mechanical Construction of Hyperbolas. — The two components of this short-circuit admittance are of opposite alge- braic sign to, but the same numerically as the two coordinates of the center of the curves in Figs. 74 and 76. Therefore, if the complex expression for this short-circuit admittance taken nega- CASE II 255 A pY tively, — -jt = —\-y coth VI, be laid off as a line on the curve sheets on which g and 6 are to be plotted in the same manner as a vector is drawn in a vector diagram, the center of the curves is at 60 \C/ B f \ \ \ g ° Fig. 77. — Graphical Construction of Curves Shown in Fig. 76. the end of the line. The circles for the first case may then be drawn according to equation (25), while the hyperbolas which give the solution for this second case may be drawn easily by a graphical process. 256 VOLTAGE REGULATION Through the curve center, located as above, or by equation (41), draw the straight lines which are the asymptotes of the hyperbola according to equation (43) for any particular value of R. In drawing these asymptotes, it is only necessary to locate one point on each line in addition to the center, C, and this is very easily done by using the known line slope, ± , Account v 2 R + R 2 must be taken, of course, of the difference in the go and b scales, ye R > \\\ O.L'l l V \\\ III 0.20 /// \\\ //'' \i o.u; / / // 0.12 1 / j / 0.08 / 1 1 MM v 10 - '; - i - 2 l i : I 1 l ) 12 Fig. 78. — Relation between 6o and R for Different Values of 0o Cross-plotted from Fig. 76. Reading from Axis of Curves Outward, for go = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 X 10" 3 mhos. if such a difference exists. By equation (42), locate also the vertex of the hyperbola; Fig. 77 shows the graphical construction of these curves, which, for convenience in reading data, should be drawn immediately upon cross section paper. O is the origin of coordinates, deter- mined by the intersection of the g and 6 axes. C is the curve center, at the end of the line representing the quantity — r 5 , the short-circuit admittance taken negatively. The point A being at the vertex of the hyperbola, the distance BA is R times the dis- case in 257 tance CB, according to equation (42). After having drawn the asymptotes and located the point A for any value of R in ques- tion, draw any line, QS, through the vertex A, and lay off the distance SP equal to AQ. The point P is then a point on the desired hyperbola. Continue in this way until a sufficient num- ber of points are obtained through which to draw the desired curve. This graphical construction remains valid even though the g and 60 scales are unequal, provided the points C and A are prop- erly located, and the asymptotes drawn according to equation (43). The family of curves in Fig. 76 could have been drawn by the above method instead of from the numerical data derived by com- putation from equation (39) and given in Table XXXIV. If the accuracy of plotted curves is sufficient, results may be obtained very rapidly by the use of this graphical process. By replotting from Fig. 76, the curves shown in Fig. 78 are obtained. Case EI. A Portion Only of the Load-end Susceptance Constant 167. Load-end Voltages. — This case is a continuation in more general form of Case II. Here, the total load end admittance is divided into two parts — the load proper consisting of g and W, b ' which make up a load whose power-factor angle is <£o = tan -1 —, do and a constant load-end susceptance, &o, which does not disappear simultaneously with the load proper, and which represents ap- proximately a synchronous condenser. The voltage regulation under these conditions is determined by keeping the generator voltage, e, constant in magnitude, as well as b 0} and noting the variation in eo when g and &</ vanish. Following the same method as in Case II: Under load, Y = go + j (bo + W), Eo = a, + jo, + (go + jbo + jW) (bx + #5 ' ^ 7) Under no-load conditions, Y ' = + jb , the load-end voltage is Eo ' = a 1 +ja 2 -{-(0-{-jbo)(b 1 +jb 2 ) ' (48) 258 VOLTAGE REGULATION If R = voltage regulation, e 2 eo" = (1 + R) 2 eo' = fa-bM+fa + bJW (49) (a x + 6,0o - kW - k&o) 2 + («2 + 6200 + biV + V>o) 2 ' Simplifying according to the method used with equations (30), (31), and (32), where m = 2 R + R 2 , -mW + 2 [&V + m (aib 2 - a*f)i)] 60 + &W + 2 (a^x + 0262) 0o + & V 2 - 2 (ai& 2 - 0261) 60' - ma 2 = 0, giving -m&W + 2 [b*W + mAxB]b + b 2 g 2 + 2A-Bg + WW 2 -2AX BW -ma 2 = 0. (50) 168. Constant Portion of Load-end Susceptance for Given Regulation. — Solving the above for b Q , the constant amount of load-end susceptance required to give a regulation R with a load of conductance g and susceptance 60', . W . A x B bo — u — m b 2 ^V (1 + m) W 2 + mg 2 + 2m^ O - m 2 ^^ 1(6V ± V/^ W 2 + i (b 2 g + A . fl)» - (l + 1) (A • *)' J • (51) In equation (51), 6 ' is the susceptance of the load proper, and when in this expression b ' is zero, the solution for b , the suscept- ance inserted for purposes of voltage control, corresponds with the solution given in equation (34) for 6 under such a special condition. As it stands, equation (51) involves two independent variables, 0o and W, which specify the load conditions, and therefore it is impracticable to plot a sufficient number of curves to cover all of the possibilities therein. If, in any particular case, the value of W is known for every value of 00 (as, for example, with a load of constant power-factor, where W = 00 tan <fo), the labor involved in computing a set of data from CASE III 259 which to plot curves similar to those in Fig. 76 is not excessive. The condition of constant power-factor of load proper will be taken up later. 169. Nature and Location of Curves. — Equation (51) assumes the form which is the equation of an hyperbola in the variables g and b when &</ is constant. The coordinates of the center of the hyper- bola are A ' B * n A V , A XB "JT and 2R + & + -V~-' (53) which are the same as given in equation (41) for a non-inductive load, with the exception of the amount „ °. D „ , which, in this Z K t ti case, is added to the & coordinate of the central point. Changing the value of 6 ' thus causes the centers of the curves to take dif- ferent positions along the perpendicular line passing through the end of the vector — ^ . The displacement of the center from this D point is 2 R + # 2 ' ^ The distance from the center to the vertex of the hyperbola is aW(nr) 2 -2TO' < 55 > and, as in the preceding case, the asymptotes to the curve have slopes ± , * (56) V2R + R 2 170. Mechanical Construction of the Curves. — Equations (54), (55), and (56) enable the hyperbolas to be plotted by the convenient graphical construction for any fixed values of R and W, though in this case the center of the curve changes its position with each change in the above quantities, according to equations (53) and (54). 200 VOLTAGE REGULATION The value of the radical in equation (55) may conveniently be found from the simple right-angled triangle relation. Draw two lines intersecting at right angles, and from their intersection lay off W along one of them, and with a radius A-B describe V2R + K 2 an arc intersecting the other. The intercept is then the desired value of the radical. Fig. 79 shows the relations existing in the diagram giving b as a function of go, for fixed values of R and &</• Having the asymptotes (^^^M Fig. 79. — Graphical Construction for Plotting the Hyperbolas Represented by Equations (51) and (52). and the vertex of the hyperbola as obtained by the construction indicated in this figure, the curve may be completed by the me- chanical process previously described, Fig. 77. In any given case, the most convenient procedure probably would be to plot on one curve sheet a number of such curves for different values of b ' = susceptance of load proper, but all for a fixed value of regulation R. For all of the curves, the asymptotes would then have equal slopes — the center and vertex only chang- ing with changes in b '. By cross-plotting from a number of such CASE III 261 curve sheets, each of which is constructed for a different value of R, almost any desired information may be obtained. Fig. 80 shows such a set of curves applying to the same transmission line as treated by Fig. 76, for the particular case, R = 0.16. The hyperbolas for &</ = 0.0006, 0.0004, 0.000, -0.0004 and -0.0006 are drawn. The curve for & ' = 0.000 is the same as that shown in Fig. 76 for R = 0.16, the difference in their appearance arising from a difference in the scales used in the two curves. *>o xl( ' 1 / / ' ■ / / 4.S A / / / / 4.0 / f — f- -" -7 — — — u 8,4 \\ V _ l.fi /i \ '/ — i _. , i i O 3 10 ! -1 (i h sV ).S ' Ai B 2 i 8 2 -0.8 -1.0 \ \ \ Axes for 0.6 x 10 s 0.4 x 10 s ■0.4 X 10 s 0.6. r^lO 8 Fig. 80. — Relation between g and b for R = 0.16 and for Different Values of W. Locate the Different Curves by Their Axes, which are Numbered with the Corresponding Values of &</• 171. Forms of Curves in Special Cases. '— As is to be noted from Fig. 80, the hyperbolas have greater curvatures at their vertices for the larger numerical values of &</• By equation (55), when &o /2 =(2# + fi 2 )(^J, (57) 262 VOLTAGE REGULATION the distance from the center to the vertex of the hyperbola is zero, and therefore the curve reduces to two intersecting straight lines — the asymptotes. For this particular case, the bo coordinate of the center of the curve is ^±^=^£ (58) from equation (53), and according to whether W is condensive or inductive the algebraic sign in (58) is positive or negative. For values of 6 ' greater than that indicated by equation (57) the right-hand member of equation (52) becomes negative, and the equation thereby represents hyperbolas whose major axes coincide with the line passing perpendicularly through the end of the vector — ■= . The centers of the hyperbolas are located as before, according to equation (53), but the vertices, now falling upon the perpendicular line through the center, are at distances 1 + R 2R + R* \Jw*-(2R + R*)(^>j (59) from the central points. The right-hand sides of both branches of these hyperbolas must be drawn in order to furnish complete in- formation by means of the double value of b secured for each value of 0o, while in the case of the hyperbolas whose major axes are horizontal, the double value of 6 is secured from both sides of the right-hand branches. In the particular numerical problem previously used for illus- A tration, where - -g = - 0.0010661 +J 0.0021141, by equation (57) the hyperbolas reduce to straight lines for W = ± 0.0010661 V2 R + R?. Taking R = 0.16 as in Fig. 80, W = ± 0.000627 mho for such a condition. In Fig. 81 the curves for W = 0.000, 0.0004, 0.000627, and 0.0010 are shown. 172. Zero Voltage Regulation. — In this case — constant susceptance of load proper — a condition of zero voltage regula- tion can be secured as long as &</ has a finite value. The graphical process of plotting the curves cannot be applied for 72 = because the centers of the curves fall at an infinite distance above the go CASE III 263 axis and the asymptotes have slopes of ± infinity. Introducing into equation (50) the condition that m = 2 R + R? = 0, and solving for &</, .1 = _|ft=0 AXB W A • B g fr 2 W 2 6 (60) \ bx io :! Jl2 ) 10 8 J 6 A xes for ft' A /— — — — K310C I 0.00062 "qooo4o[i* It A (/o ' io 3 f i 1 \ 2 4 3 10 12 S" 2 _j i -4 ■ -0 1 Fig. 81. — Relation between g and b for R = 0.16 and Different Values of 6 '. The Curves Become Straight Lines for 6 ' = 0.000627. the equation of a parabola in g and 6o- Numerical data for curve plotting is so easily obtained from the above equation that graph- ical methods of construction need not be developed. For every real value of g as well as for every real value of &</ a real solution for 6 is obtained. No imaginary solutions, indicating impossible physical conditions, are to be obtained. In general, when negative values] of R are considered, corre- sponding to a decrease in load-end voltage when the load proper 264 VOLTAGE REGULATION is removed, it is easily seen by equation (52) that the curve re- lating 0o and &o is an ellipse — throughout certain ranges in the value of R. Being of lesser ^importance, detailed development of this matter is not given here. Case IV. The Load Proper of Constant Power-Factor 173. Equation Relating Constant Susceptance and Regula- tion. — Under this head is considered a load proper the two components of whose admittance are always in a constant ratio to each other. The power-factor angle of the load is then constant, and the problem is to determine the proper constant amount of additional susceptance at the load end to produce certain values of voltage regulation. Let 0o = power-factor angle of load proper, counted positive for a leading load. k = tan 0o = — , W = kg . 0o (61) go and b r are the two components of the load admittance. The above value of W may be substituted directly in equation (50), giving (1 + V) gfo 2 + 2 kgobo - m6o 2 + 2 (^ - k ^jffi ft + 2m^J*-«^0. (62) 174. Location of Curves. — (62) is a general equation of the second degree in g and & as variables, and it therefore represents as a curve some conic section whose center is displaced from the origin of coordinates and whose axis is angularly displaced from the coordinate axes. From the coefficients of this equation and by the methods of analytical geometry we find that the major axis of the conic section or curve makes an angle, a, with the g axis (counted counter clockwise) determined by tan 2 « = 1+ l*+A.' a = \^- l JT+W+¥' (63) CASE IV 265 The center of the curve is displaced from the origin of coordinates by the amounts 2 R + R 2 A-B and (1 + R) 2 k 2 + 2R + R 2 AxB k 6 2 A>B b 2 (64) b 2 (l + R) 2 k 2 + 2R + R 2 along the g and 6 axes, respectively. 175. Nature of Curves. — The nature of the curve represent ing equation (62) is determined as follows: < 0, an ellipse, (1 + R) 2 k 2 + 2 R + R 2 = 0, a parabola, > 0, an hyperbola. (65) Since negative values of R are seldom considered and since k enters into these determining conditions only as k 2 , the left- hand member in the expression (65) is usually positive and greater than zero. The curve is then usually an hyperbola, and graphical methods of construction of the curve for this case only will be considered. To determine the slopes of the asymptotes to the hyperbola it is only necessary to determine the slopes of the curve represented by equation (62) at infinity. Since g and 6 have infinite values at an infinite distance from the origin, the first powers of these variables as well as the constant term in the equation (62) may be neglected in comparison with their squares and product. Thus, as approximation at infinity, (l+fc 2 )0o 2 +2fc<7o&o-m&o 2 = O, <7o 2 + 7^5 9oh - r^T5 &o 2 = 0. (66) 1 + k 2 1 + k 2 Assuming this equation to consist of the product of two linear factors, as (go + X&o) (00 + yb Q ) = 0, 1 go 2 + (X + y) gob + X7&0 2 = 0, J K } and equating coefficients of like terms in equations (66) and (67), X = 7 = 1c + Vk 2 + m (1 + k 2 ) l + k 2 Vk 2 + m (1 + A; 2 ) 1 + & 2 (68) 266 VOLTAGE REGULATION From the two equations, go + X&o = and g + 7&o = 0, and the relation, m = 2 R + R 2 , the slopes of the asymptotes are determined as l + fc» Slopes of asymptotes = — k d= "v kt -f m ^i -j- «*; i _i_ 1.2 (69) A; ± VK* + m (1 + k 2 ) 1 + fc 2 fc ± V ( 1 + R) 2 k 2 + 2 # + it! 2 176. Mechanical Construction. — Knowing, from equation (64), the position of the center of the curve and from equation (69) the slopes of the asymptotes, the hyperbola is completely determined if the position of one point on the curve is known. Two such points may be determined by placing 6 = in equation (62) and solving for g . Thus, for 6 = 0, ±V(A -B-kAxB) 2 + m{\ + k 2 ) a 2 b 2 \. (70) From the data in equations (63), (64), (69), and (70), the hyper- bola may be constructed by the graphical or mechanical process described in Fig. 77, and the labor involved is not excessive. In this connection it is well to note that the axis and asymptotes should be drawn first, for then it may be possible to determine by inspection whether the two points on the curve, as located by equation (70), will be suitably situated for an accurate continuation of the mechanical process of curve construction. If the two points so located should happen to fall very near to the asymp- totes (as compared with the distances from the points to the center) accurate construction by the previously described method is not readily obtained. In such cases, a value of 6 equal approxi- mately to the ordinate of the center of the curves may be substi- tuted in equation (62) and the corresponding values of go deter- mined by solving the resulting quadratic equation. These points will generally fall near the vertices of the curves — the ideal condition as regards the accuracy of the graphical process of curve drawing. CASE IV 267 177. Numerical Illustration. — Figure 82 shows the hyperbola giving the relation between g and b Q for the numerical case under discussion, for the arbitrarily selected conditions: R = 0.16 and k = tan <£o = —0.75, which corresponds to a lagging load of 0.80 \\ M 10 3 \ \ \ 6 \ /, ^ \ 5 ^ / \ J // ^ \ 4 ^ "•- ^■^ I x ""-, \ / ,/ T *--y '/ / \ ■^^ ■~. // V \ // 1 A \ I \ (/o> : 10 i -• / 3 - 2 1 °\\ i 2 3 / -1 \ \ \ -2 i \ -3 \ Fig. 82. — Load-end Snsceptance b Required for a Regulation of 0.16 at a Constant Power-Factor of Load of 0.80, Lagging. power-factor. The data from which the curves may be constructed are: Angle, a, between axis of curves and go axis = —19.086 deg. Angles between asymptotes of curves and g axis, +40.959 deg. and -79.131 deg. Coordinates of center, -0.0003342 and +0.002839. 0o for 6 = 0, -0.0037268 and +0.0003327. g for 6 = 0.00300, -0.0009578 and +0.0004437. In this particular case the positions of the points as determined for 6o = were not suitable for the process of graphical con- 268 'VOLTAGE REGULATION struction. The positions of the two points for 6 = 0.00300 were therefore computed, and the points so located used as a basis for further construction. The foregoing methods for the determination and construction of voltage regulation curves for transmission systems are merely illustrative of general methods which may be employed. In regard to the determination of the proper load-end susceptance to give specified values of regulation, a number of special condi- tions have been considered, but it is obviously impossible to take up individually the great number of such conditions which can arise. In a subsequent volume, dealing primarily with power transmission, a more complete discussion is contemplated. It is obvious that similar developments covering the transmission of power by means of constant, or nearly constant, current are possible. APPENDIX HYPERBOLIC FUNCTIONS A description of the hyperbolic functions with their geometrical significance is not required in order to lead to a clear understanding of their use in the analytical expressions of the solutions of physical problems. For such purposes, the definitions . e x + e~ x .x 2 x* . coshx=— 2— = 1+2+24+ ••' , pX __ p — X yjr»3 /v»5 sinhz =—£- - = ^+6 + 120 + , sinhz x 3 . 2x 5 tanh x = — r— = x — «" + te" r coshz 3 15 (1) usually suffice. Familiarity with the types of differential equa- tions whose solutions are expressible in terms of these functions is desirable. Fig. 83 shows the curves representing the hyper- bolic functions of a real variable, and as is apparent from an inspection of the series for these functions, their nature is essentially more simple than that of the trigonometric functions, since they are not periodic. An excellent working table o the natural values and logarithms of these functions is given by Becker and Van Orstrand in the "Smithsonian Mathematical Tables." The values are given to five decimal places. A collection of formulae relating to the hyperbolic functions is also given by them, and on account of its usefulness it is, by the permission of the publishers, reproduced here in part. A very useful table of the hyperbolic functions of complex variables has recently been published by Professor A. E. Kennelly. It is particularly useful in the rapid determination of approximate solutions, but for very precise work, the double interpolations involved render its application cumbersome. As before stated, a five-place accuracy in the tabulated values of functions is sufficient for the majority of physical and engineer- ing calculations, but since a table of hyperbolic functions is not 269 270 APPENDIX used as frequently as tables of trigonometric functions or common logarithms, it appeared desirable to the writer to prepare and publish herewith a six-place table of the logarithms of hyperbolic functions. They provide for the exceptional case which requires the degree of precision afforded by logarithmic computations to 2.0 ].N \c iosii X l.fi 1.4 u 1.0 Sin l X .8 Ta ih^r .fi .1 .2 -1 ;-i 2-1 o -. B - 6 - i -. 1 / > . 1 . r, >s 1 1 2 I 4 -A -.0 Ta ill 1 -s -1.0 ' -1.2 -l.f Sin' 1 X -l.G Fig. 83. — The Hyperbolic Functions of a Real Variable. six significant figures, and at the same time suffice, without an undue increase in the labor of interpolation, for five-place com- putations. The six-place tables which follow were taken from twelve- place tables which were computed by B. M. Woods and the writer, and which appear in the Engineering Series of Publications of the University of California. Tabulations of logio cosh x, logio sinh x, logio tanh x, logio , and logio 1 — r— are given for APPENDIX 271 values of x from 0.000 to 2.000, which range is ample for the greater portion of engineering problems of the nature treated in this work. Tabular intervals of 0.001 in the argument and the x render interpolation sinh x use of the quantities log and log tanhx easy in all portions of the table. SI 111 1 jC The quantities log - ; — and log are given for values of x ~ tanh x x from 0.000 to 0.500, and serve for the determination of log sinh x and log tanh x throughout this range where otherwise the inter- polations would be troublesome. Thus, given log x and x: sinhx and log sinh x = log x + log log tanh x = log x — log tanhx (2) Given log sinh x or log tanh x, interpolations by proportional x sinh x parts may be made for log — - — and log x " tanh x differences for the latter quantities are small. Thus, sinhz , since the tabular and log x = log sinh x — log : log x = log tanh x + log tanhx (3) The method of use of the auxiliary tables is identical with that of the "S and T" tables used for the determination of the loga- rithms of the trigonometric sines and tangents of small angles. In the portion of the table from x = 0.500 to x = 2.000, the differences between successive values are tabulated, and it is to be noted that these differences are such that linear interpolations only are required. 272 APPENDIX Relationships Involving Hyperbolic and Allied Functions Reproduced by permission from the Smithsonian Mathematical Tables A. — Relations between Hyperbolic and Circular Functions 1. sinh u = —i sin iu = tan gd u. 2. cosh u = cos iu = sec gd u. 3. tanh u = — i tan iu = sin gd u. 4. tanh \ u = tan \ gd u. 5. e" = (1 + sin gd u) + cos gd u = [1 — cos (| 7r + gd u)] + sin (£ 7r + </d w) = tan (j 7r + I gd u). 6. sinh iu = i sin u. 7. cosh iu = cos u. 8. tanh iu = i tan u. 9. sinh (u ± £y) = ± i sin (y =F iu) = sinh it cos v ± i cosh m sin y. 10. cosh (u ± iv) = cos (y Tiw) = cosh u cos y ± i sinh u sin y. 11. cosh (miir) = cos m*-. (m is an integer.) 12. sinh (2 m + 1) \ in = i sin (2 m + 1) | x. (w is an integer.) B. — Relations among the Hyperbolic Functions 13. sinhu = \ (e u — e - ") = —sinh ( — u) = (cschu) -1 = 2 tanh | m 4- (1— tanh 2 \u) = tanh u 4- (1— tanh 2 u)*. 14. cosh u = \ (e u + e~") = cosh (— m) = (sech w) -1 = (1 + tanh4u) 4- (l-tanh^u) = 1 4-(l-tanh 2 u)*. 15. tanh u = (e u - e~ u ) 4- (e u + e - ") = —tanh (— u) = (coth m) _1 = sinh u 4- cosh w = (1 — sech 2 u)^. 16. sech u = sech (— u) = (1 — tanh 2 u) 5 . 17. cschu = — csch (— u) = (coth 2 u — 1)^. 18. coth u = —coth (— u) = (csch 2 u -\- 1)K 19. cosh 2 u — sinh 2 u = 1. 20. sinh £ u = V^ (cosh u — 1). 21. cosh |u= V^ (cosh w + 1). 22. tanh \u = (cosh u — 1) 4- sinh u = sinh u-7- (1+cosh u) = \/(cosh w— 1) 4- (cosh w+1). APPENDIX 273 23. sinh 2 u = 2 sinh u cosh u = 2 tanh u-f (1- tanh 2 w). 24. cosh 2w = cosh 2 w + sinh 2 u = 2 cosh 2 w — 1 = 1 + 2 sinh 2 u = (1 + tanh 2 u) 4- (1 - tanh 2 u). 25. tanh 2 w = 2 tanh w + (1 + tanh 2 u). 26. sinh 3 u = 3 sinh u + 4 sinh 3 u. 27. cosh.3 w = 4 cosh 3 w — 3 cosh u. 28. tanh 3 u = (3 tanh u + tanh 3 u) 4- (1 + 3 tanh 2 u). 29. sinh mt = ii-i . (n) (n— 1) (n — 2) , , . , _ n cosh n_1 m sinh u+ —^ -pr-^ cosh" -3 u sinh 3 u -f- • • • . 6 30. cosh nu — cosh" u -\ *-= cosh 71-2 w sinh 2 it 4> • • • . 31. sinh w + sinh v = 2 sinh | (it + y) cosh § (w — y). 32. sinh it — sinh v = 2 cosh | (w + w) sinh ^ (m — y). 33. cosh it + cosh v = 2 cosh ^ (w + y) cosh \ (u — v). 34. cosh u — cosh v = 2 sinh | (it + y) sinh | (u — v). 35. sinh u + cosh it = (1 + tanh J u) 4- (1 — tanh £ it). 36. (sinh it + cosh u) n = cosh nw + sinh ww. 37. tanh u + tanh y = sinh (u + y) 4- cosh it cosh y. 38. tanh u — tanh y = sinh (it — v) 4- cosh w cosh y. 39. coth u + coth y = sinh (it + y) + sinh it sinh y. 40. coth it — coth y = — sinh (w — y) 4- sinh w sinh v, 41. sinh (it ± y) = sinh u cosh y ± cosh u sinh y. 42. cosh (u ± y) = cosh w cosh y ± sinh u sinh y. 43. tanh (it ± y) = (tanh w ± tanh y) 4- (1 ± tanh w tanh y). 44. coth (it ± y) = (coth w coth v ± 1) 4- (coth y ± coth it). 45. sinh (it + y) + sinh (u — v) = 2 sinh it cosh y. 46. sinh (it + y) — sinh (w — y) = 2 cosh u sinh y. 47. cosh (u -f- y) + cosh (it — y) = 2 cosh u cosh y. 48. cosh (it 4> y) — cosh (it — y) = 2 sinh it sinh y. 49. tanh \ (it + y) = (sinh w + sinh y) 4- (cosh it + cosh y). 50. tanh \ (it — v) = (sinh u — sinh y) 4- (cosh u + cosh y). 51. coth I (w + v) = (sinh it — sinh y) 4- (cosh u — cosh y). 52. coth \ (u — y) = (sinh u + sinh y) 4- (cosh m — cosh y). _„ tanh m + tanh y _ sinh (u + y) tanh m — tanh y sinh (w — v) K . coth u + coth y _ _ sinh (u + y) coth u — coth y sinh (u — y) 55. sinh (u + y) + cosh (u + y) = (cosh u + sinh w) (cosh y + sinh y) . 274 APPENDIX 56. sinh (u -f v) sinh (u — v) = sinh 2 u — sinh 8 v = cosh 2 u — cosh 2 v. 57. cosh (it -+- v) cosh (u — v) = cosh 2 u + sinh 2 v = sinh 2 u + cosh 2 v. 58. sinh (miV) = 0. (m is an integer.) 59. cosh {mix) = ( — l) m . 60. tanh (miir) = 0. 61. sinh (u -f- mzV) = (— l) m sinh u. 62. cosh, (u -f- rraV) = ( — l) m cosh u. 63. sinh (2m+l)} Mr = ±t\ 64. cosh (2 m + 1) \ Mr = 0. 65. sinh f y ± u J = t cosh u. 66. cosh [ y ± w) = ±i sinh u. 67. tanh (u + tV) = tanh u. C. — Inverse Hyperbolic Functions 68. sinh- 1 u = log (if +Vu 2 + l) = cosh" 1 Vi?+1 = f — ^— , • ^ (u 2 + l)* 69. cosh- 1 u = log (u + Vu 2 - l) - sinh- 1 v^l = C—^L— . J (u 2 -l)* 70. tanh" 1 u = \ log (1 + u) -\ log (1 - u) = fj^' 71. coth- 1 M = ^log(l + W )-^log(M-l)= f-^- 2 = tanh" 1 -- J 1 — ir u 72. sech- 1 M=logf-+v/-\-l) = - f ^— ^cosh- 1 ^ V« y w 2 / J u(l -u 2 )* u 73. csch- 1 u=log(- + V^ + l) = - f — r = sinh" 1 -- 74. sin -1 u = — isinh -1 iu = — i log (iu + Vl — u 2 ). 75. cos -1 u = — i cosh -1 u = — i log (u + i Vl,— u 2 ) . 76. tan -1 u = — i tanh -1 iu = 75-. log (1 + w) — —. log (1 — iu). 77. cot -1 u = i coth -1 iu = =-? log (iu — 1) — 7r . log (iu + 1). 78. sin -1 iu = i sinh -1 u = i log (u + Vl -f- u 2 ). APPENDIX 275 79. cos -1 iu = — i cosh -1 iu = g — i log (m + vl + w 2 ). i % 80. tan -1 iu = i tanh -1 u = = log (1 + w) — ~ log (1 — w). 81. cot -1 iu = —i coth -1 w = — 5 log (u + 1) + 5 log (w — 1). 82. cosh" 1 h(u + i) = sinh" 1 § ( w - ^j = tanh- 1 grjOJ = 2 tanh -1 — r-= = log w. M + 1 83. tanh -1 tan u — \gd2u. 84. tan^ 1 tanh u = \ gd~ l 2 w. 85. cosh -1 esc 2 m = — sinh -1 cot 2 w = — tanh -1 cos 2 w = log tan u. 86. tanh -1 tan 2 ( j x + \ u) = \ log esc w. 87. tanh -1 tan 2 % u = \ log sec it. 88. cosh -1 u ± cosh -1 v = cosh -1 [uv ± V(u 2 — 1) (v 2 — 1)]. 89. sinh" 1 u ± sinh" 1 = sinh" 1 [u Vl + y 2 =b i> vT+m 2 ]. D. — Series AA ^ 1 . 14/ . 14 . 14/ . 90. e u = l+" + 2! + 3l + 4l+ '"• 91. logu=(M-l)-|(u-l) 2 + i(M-l) 3 - • • • . »^^ + "lP=i)r + ip=i)r + ... 94. log (1 + u) = u — g M 2 + ~ w 3 — 7 M 4 + • • • . 95. log(j^)=2[ M + | W 3 + jU 5 + ^u 7 + • • -J. w 3 u^ u} 97. sinh m = m + 31 + 5"j + t]+ ' * * = "( 1 ^)( 1+ p)( 1 + 3^)---; 98. coshw=l+^ + ^ + |-j+ • .- (m 2 < 00 . (2>m>0. . (u > \. 1. (m>0. (m 2 < 1. (u 2 < 1. (u 2 > 1. (u 2 < 00 . (u 2 < 00 . (u z < 00 . 276 APPENDIX "*" 3^7 V 1 + 5*^J hmhmhm 99. tanh u = a- s u 5 + — «* - L5 u- V7_ 315 u 7 + 100. u coth u=l+„-u ! -^,u 4 +J;« 6 - 3 45 94d 101. sech u=l-n« ! + 21 u l 61 720 u 6 + 31 102. M cschu=l-^u^3^^- 1 u 6 + 103. ^^ = <^ = ^-6" 3 + 24" 5 -5oTo w7 + x , 1 sech 3 m 13 sech 6 it = __ sech u-2-3 24-5-- 104. i« = flK*- l * = * + |*» + ^ 61 5040 <*> 7 + in _ . , . 1 u 3 , 1 3 m 5 1 3 5 m 7 , 105. ^ 1 « = «- 2 3+245-2467 + -' 1 1 13 1 135 1 g + 22m 2 244u 4 + 2466u 6 1M ._, . 11 13 1 135 1 106. cosh ^=1082^-2^-244^-2466^- 107. tanh- 1 ** = u + 5 u* + \ u* + \ u 7 + • • • . 6 7 108. coth- 1 u=tanh- 1 - = - + ^ 3 + ^- 5 + ; ^-:+ • . u u 3u 3 5 m 5 7w 7 109. sech iu=cosh i- = l g-----244 -^ee " (u 2 < 00 .) (m 2 <1t«. (M 2 < X 2 . (W 2 <7T 2 . (m small.] . (u large. (u 2 < 10 . (u 2 >l. •• (w 2 >l. (u 2 < 1. . (u 2 > 1. lia. csch -1 u=sinh- 1 - = ^5— -f ~t-=— - 6 — st a^t^f "+ u u 23u 3 245 w 6 2467 m 7 (m 2 < 1. = 1 2, 1m 2 _13m| 135«;_ g u + 2 2 244 + 2466 (w 2 > 1. . (w 2 <l.) E. — Derivatives 111. ^- = e\ du 112. d lo&iM du u da v dv , 113 - iH =a -Tu- log ' a - 114. ^ =««(! + log. «). APPENDIX 277 , ., _ d sinh u , . ., „ d tanh u , „ 115. — j = coshw. 117. 5 = sech 2 M. du du „.,- d cosh w . . ,„ rf coth u ,. 116. ; = sinnw. 118. 5 = — csch 2 w. du du 119. — -i = — sechw. tanhw. du 120. — -, = — cschu. cothw. du 101 d sinh -1 u _ 1 10K dsech _1 M <*U V« 2 + 1 du u y/l ^ nn d cosh -1 u 1 , M d csch -1 w 122. ; = ■ ; 12b. du Vu 2 - 1 ^w w Vu 2 + 1 nh" 1 m = 1 127 rfgdi du 1 — u 2 du >th -1 u = 1 12g rfgd- dw 1 — w 2 ' du 00 dtanh 1 !* 1 10 _ dgdw , 123. j = : =• 127. —3 — = sechw. du 1 — u 2 du 10 . dcoth _1 M 1 100 dgd _1 w 124. ; = ; 5- 128. -~ = sec u. F. — Integrals. (Integration constants are OMITTED.) 129. I sinh udu = cosh u. 131. I tanh udu = log cosh u. 130. / cosh u du = sinh w. 132. I coth udu = log sinh u. 133. / sech udu = 2 tan -1 e" = gd u. 135:. / csch udu = log tanh 5 • 135. / sinh n u du = - sinh n_1 w. cosh u / sinh 71-2 u du J n n» J = — r-s sinh n+1 u cosh u f-s / sinh n+2 u du. n + 1 w+ 1 J 136. / cosh n u dw = - sinh u. cosh n_1 u -\ / cosh" -2 u du J n n J 1 n + 2 C = r-r sinh u cosh n+1 u H :— 7 I cosh n+2 u du. , n+ 1 n+ 1 J 137. / m sinh udu = u cosh w — sinh u. 138. / m cosh udu = u sinh w — cosh it. 278 APPENDIX 39. I m 2 sinh udu = (u 2 + 2) cosh u — 2 u sinh u. 40. / u n sinh udu = u n cosh u — nu n ~ l sinh w + n (n — 1) / u n- a sinh u dw. 41. / sinh 2 u du = \ (sinh u cosh u — u). 42. I sinh u. cosh u du = \ cosh (2 u). 43. / cosh 2 u du = \ (sinh u cosh u + u). 44. I tanh 2 udu = u — tanh u. '45. I coth 2 udu = u — coth u. 46. / sech 2 udu = tanh u. 47. / sech 3 udu = \ sech u tanh u -}- $ gd u. . j csch 2 udu = — coth u. I sinh -1 udu = u sinh -1 u — (1 + u 2 )*. . / cosh -1 udu = u cosh -1 u — (u 2 — 1)*. 51. I tanh -1 udu = u tanh -1 u + \ log (1 — u 2 ). 52. J u sinh -1 u du - | [(2 u 2 + 1) sinh -1 u - u (1 + u 2 )*]. 53. fu cosh -1 u du - I [(2 u 2 - 1) cosh -1 u-u(u 2 - 1)*]. 54. / (cosh a+ cosh u) -1 du = 2 csch a. tanh -1 (tanh £ u. tanh \ a) = csch a [log cosh | (w + o) - log cosh \{u — a)]. 55. / (cos a -f cosh u) -1 du = 2 esc a. tan -1 (tanh £ u. tan \ a). 56. / (1 + cos a. cosh u) -1 du = 2 esc a. tanh -1 (tanh \ u. tan \ a). APPENDIX 279 157. / sinh u cos u du = \ (cosh u. cos u + sinh u. sin u). 158. / cosh u. cos udu = \ (sinh u. cos w + cosh u. sin u). 159. / sinh u. sin udu = \ (cosh w. sin u — sinh w. cos u). 160. / cosh m. sin u du = | (sinh w. sin w — cosh u. cos u). 161. / sinh (mu) sinh (nw) du -j— — ^ [m sinh (nw) cosh (mw) — n cosh (nit) sinh (mw)]. mf — n" 162. I cosh {mu) sinh (nw) dw — — — - 2 [w sinh (nw) sinh (mu) — n cosh (nw) cosh (mu)]. 163. / cosh (mu) cosh (nu) du 2 — — 2 [m sinh (mu) cosh (nw) — n sinh (nit) cosh (raw) 1 . wr — rr 164. / sinh u tanh udu = sinh u — gdu. 165. I cosh m coth udu = cosh w + log tanh ^ • 166. I sec it dw = gd _1 u. 167. lsec 3 0d</>= / (1 + tan 2 0)^dtan<£ = ^sec0tan<^ + |gd~ '</> = \ tan 0(1 + tan 2 <t>) a + ? sinh -1 (tan <£). Here tf> = gd u. ico C du . . ,u f* du . ,u 168. I r = sinh * -• I r = sin -1 -• •/ (u 2 + a 2 )* a J (a 2 -u 2 )? a 189. r_^L_- = cosh-^. r_^^_=co S -^. J (u 2 - a 2 ) 5 a J (a 2 - w 2 ) 5 a 170. f ,, d \ -I**** f^,= lta^. J (a 2 — w 2 ) u<a a a J a? -{- u 2 a a J (ir — a 2 ) u>a a a J a —du 1 , i w - = -cot -1 — 2 + m 2 a a 172 . f ~ d » , -iwelr*". f ^—.-iwc-tH. 280 APPENDIX 173. f - dU . -leech-'?. f ~ d " , -JMB-a. »/u(a 2 + u 2 )' a a J M (u 2 -a 2 )* a o ii a C du 1 . , , au + i , 174. I r = — ■ t=. sinh _1 j- (a positive, ac> ft 2 .) J (au 2 +2 6u+c)* v^ (ac-ft 2 )* ^ »w<^«r.j 1 ,_, au + b , = — 7= cosh l -r (a positive, ac< ft 2 .) vo (b 2 -ac)* i •,,. _i_ *» 1 _, au 4- 6 / x . v 7== cos Ti; a* ( a ne 8 atlve -) —a (6 2 — acy v -a (ft* - acp 17 e r du 1 aw -f b . . ., . 175 - J(au 2 + 26u + c) = ^^ tan (^3^)1 <">*> = — ^tanh-i-^ ( ac<62 ' \ (ft 2 - ac)* (6 2 - ac)* Van + 6 < (ft 2 - ac)lj -_zL_oaai-*^±i r . ( ac<62 - ^ (ft 2 - ac)* (ft 2 - ac)* \ au + b > (ft 2 - ac)V 176 . f <fe = _2 tanh -i J«E5 f «/ (a - tt) (u - 6)* (a - 6)* V a - 6 * -2 . t /w-6 or r tan -1 V r , (6 - a)* V & - a 2 or — (a 2 ay ■ u «• — t coth -1 y _ , . (The real form is to be taken.) 177. f— life 2 ^J^ J (a - u) (6 - m)* (6 - a)* V 6 - a 2 , . . . Ib-u or t coth * V t > (6 - a)* v ft- a or j tan -1 y r . (The real form is to be taken . ) 178. T(w 2 - a 2 )* du = ^ u (u? - a 2 )*- ^ a 2 cosh" 1 -• 179. f(a 2 - u 2 )* du = ^ u (a 2 - w 2 )* + J a 2 sin" 1 -• 180. f(v? + a 2 )* du = I u (u 2 + a 2 )* + £ a 2 sinh" 1 -• J J JO 182. ue au du = ^ (au - 1). APPENDIX 281 /u m e" u m C 1 u m e? u du = 1 / u m- V u du. a aj, r e° u du _ 1 |~_ _^_ P e? u du ] 84, J vr "m-lL u*»- 1 + a J u*"- 1 ]' 85. fa bu du = ^t J bloga /a u u n na u u n ~ x , n (n — 1) a u u n ~ 2 log a (log a) 2 (log a) 3 n (n - 1) (n -2 ) . . . 2-la» /Yr. u rlii. i 87 (loga) n+1 log a (log a) 2 /ardu _ a" 1 lo u n ~ n - 1 L W -1 in - 2) n"" 2 (n-2) (n-3) n"- 3 (log a)"- 1 Ai tt dn ~] | " ' ' + (n-2) (n-3) . . . 2*1 J u \' 00 Ca u du , . (m log a) 2 (n log a) 3 88. J— = logn + nloga+-^| r + 3.3, + •••• 89. Jjfe- log j dn 1 90. C—ri— = —[mu- log (a + &e"«)]. 91 . r — *• __* tw*fr-^). J ae™ + be-" 1 " m (^i V V ft/ 92 - / / — Hi — a = 7= Dog (Va + be mu - Va) - log (Va+ 6e mu + Va)]. /■ jue»du_ e u 9d ' J (l + n) 2 ~l + u' /• . , e° u logn 1 Pe au du 94. le au logndn = 5 I J a aj u 95. / log n dn = u log n — w. 96. fu m log udu = u m+1 \^£ - , * J - J L™ + 1 (m + 1) 2 J 97. / (log n) n dn = u (log w) n — n / (log n) n_1 dn. 98. fu m (log n)» dn = um+l ^ u)n - — X7 /V (log u)*" 1 <*w. J v— » / m + 1 w + lj / (log n) n dn _ (log n) n+l n n + 1 282 APPENDIX m Ji— = log(logu) + logu + ^72! + 3T3! + "\ 201 C du 5 _J_ r du t ™ J (log u)» (n - 1) (log u)"" 1 T n-lJ (log u) n ~ l <xv> C u m du u m+l , m + 1 C u m du J (logu)" " (n - 1) (logw)"- 1 + n - 1 J (logii)— 1 ' 203. / -j = I — dy, where y = — (m + 1) log m. 204. f**_ = log (logu). J u log u 205 f—^— = - J u (logM) n (n — 1) (logu) n_1 206. f(a + bu) n log u du = s 1 |\ , , wl , f(a + 6m)"» +1 oV | F^TTT) L (a + 6m) log u ~ J i J ' 207. / u m log (a + 6m) du = _yy« * (.+*)- > /£*?]. 20g Hog (a + 6m) <fa = J u . . . 6m 1 /6m\ 2 . 1 /buV loga.log W + --2H-)+3 2 (-] , 4 (log6w)2 _ 6 ± + l(^_l(«J + .... /Mog_udu_ i r iog^ , r rfM l J (a + &»)"• b (m - 1) L (a + few)"- 1 J m (a + 6m)"- 1 J ' om Tlogudtt 1, , , i , s 1 riog(a + 6w) . 211. J {a + bu)\o g udu = ^^\o g u- a ^-au-\bu\ 212. fJ^ wdw - J (a (a + 6m) * - Va log (Va + bu — Va)], if a > 0, ^ [(log u - 2) V(a + 6m) + Valog (Va + 6m + Va) = ^[(logM-2) V(a+6M) +2 V^tan" 1 \/ ^rJ > if ° < °- APPENDIX 283 213. r e-* 1 « t du = ^ = 7 }-r(l). Jo 2a 2a \2/ 214. £ U -e-'du = T^- i l n + 1 rt»»+l a n 215. f.-^,. 1 ' 3 ' 5 -,-' 2 "-" ^. Jo 2 n+1 a n V a 217. r c-»" Vtt du « JL t/I. Jo 2 n V n 218. r e ^-du = J-- (n>0.) Jo Vm v w >f 219 220 sinh (nu) 2 n r°° udu t 2 sinh (nw) 4 w 2 Xv* nix sinh (raw) • sinh (nu) du = j cosh (raw) • cosh (nw) du = 0, if ra is different from n. cosh 2 (raw) du = — / sinh 2 (mu) du = -~ 223. / sinh (raw) du = 0. J— Mr 224. I cosh (m«) du = 0. 225. / sinh (mu) cosh (nu) dw = 0. J— »*■ 226. I sinh (raw) cosh (mu) du = 0. 227. rMi dtt= _^. Jo 1 — u 6 228. /"Mil <*„=-£. Jo 1 + m 12 284 APPENDIX 233. P(\ogu) n du= (-l)-.n!. 886. jT-^U.v?. 237. ^ tt-log (±)"du = ( ^ ( ^ ) !! 1 , if m + 1 > 0, n + 1 > 0. APPENDIX 285 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS X log cosh x log sinh x log tanh x , sinh x log X M#l iT~ tanh x 0.000 0.000 000 -Inf. -Inf. 0.000 000 0.000 000 0.001 0.000 000 7.000 000 7.000 000 0.000 000 0.000 000 0.002 0.000 001 7.301030 7.301 029 0.000 000 0.000 001 0.003 0.000 002 7.477 122 7.477 120 0.000 001 0.000 001 0.004 0.000 003 7.602 061 7.602 058 0.000 001 0.000 002 0.005 0.000 005 7.698 972 7.698 966 0.000 002 0.000 004 0.006 0.000 008 7.778 154 7.778 146 0.000 003 0.000 005 0.007 0.000 011 7.845 102 7.845 091 0.000 004 0.000 007 0.008 0.000 014 7.903 095 7.903 081 0.000 005 0.000 009 0.009 0.000 018 7.954 248 7.954 231 0.000 006 0.000 012 0.010 0.000 022 8.000 007 7.999 986 0.000 007 0.000 014 0.011 0.000 026 8.041401 8.041 375 0.000 009 0.000 018 0.012 0.000 031 8.079 192 8.079 160 0.000 010 0.000 021 0.013 0.000 037 8.113 956 8.113 919 0.000 012 0.000 024 0.014 0.000 043 8.146 142 8.146 100 0.000 014 0.000 028 0.015 0.000 049 8.176 108 8.176 059 0.000 016 0.000 033 0.016 0.000 056 8.204 139 8.204 083 0.000 019 0.000 037 0.017 0.000 063 8.230 470 8.230 407 0.000 021 0.000 042 0.018 0.000 070 8.255 296 8.255 226 0.000 023 0.000 047 0.019 0.000 078 8.278 780 8.278 701 0.000 026 0.000 052 0.020 0.000 087 8.301059 8.300 972 0.000 029 0.000 058 0.021 0.000 096 8.322 251 8.322 155 0.000 032 0.000 064 0.022 0.000 105 8.342 458 8.342 353 0.000 035 0.000 070 0.023 0.000 115 8.361 766 8.361651 0.000 038 0.000 077 0.024 0.000 125 8.380 253 8.380 128 0.000 042 0.000 083 0.025 0.000 136 8.397 985 8.397 850 0.000 045 0.000 090 0.026 0.000 147 8.415 022 8.414 876 0.000 049 0.000 098 0.027 0.000 158 8.431 417 8.431 258 0.000 053 0.000 106 0.028 0.000 170 8.447 215 8.447 045 0.000 057 0.000 113 0.029 0.000 183 8.462 459 8.462 276 0.000 061 0.000 122 0.030 0.000 195 8.477 186 8.476 991 0.000 065 0.000 130 0.031 0.000 209 8.491431 8.491223 0.000 070 0.000 139 0.032 0.000 222 8.505 224 8.505 002 0.000 074 0.000 148 0.033 0.000 236 8.518 593 8.518 356 0.000 079 0.000 158 0.034 0.000 251 8.531563 8.531 312 0.000 084 0.000 167 0.035 0.000 266 8.544 157 8.543 891 0.000 089 0.000 177 0.036 0.000 281 8.556 396 8.556 115 0.000 094 0.000 188 0.037 0.000 297 8.568 301 8.568 004 0.000 099 0.000 198 0.038 0.000 313 8.579 888 8.579 575 0.000 105 0.000 209 0.039 0.000 330 8.591 175 8.590 844 0.000 110 0.000 220 0.040 0.000 347 8.602 176 8.601 828 0.000 116 0.000 232 0.041 0.000 365 8.612 906 8.612 541 0.000 122 0.000 243 0.042 0.000 383 8.623 377 8.622 994 0.000 128 0.000 255 0.043 0.000 401 8.633 602 8.633 201 0.000 134 0.000 268 0.044 0.000 420 8.643 593 8.643 173 0.000 140 0.000 280 0.045 0.000 440 8.653 359 8.652 920 0.000 147 0.000 293 0.046 0.000 459 8.662 911 8.662 452 0.000 153 0.000 306 0.047 0.000 480 8.672 258 8.671 778 0.000 160 0.000 320 0.048 0.000 500 8.681408 8.680 908 0.000 167 0.000 333 0.049 0.000 521 8.690 370 8.689 849 0.000 174 0.000 347 286 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log cosh x log sinh x log tanh x , sinh x log X tanhz 0.050 0.000 543 8.699 151 8.698 608 0.000 181 0.000 362 0.051 0.000 565 8.707 758 8.707 194 0.000 188 0.000 376 0.052 0.000 587 8.716 199 8.715 612 0.000 196 0.000 391 0.053 0.000 610 8.724 479 8.723 869 0.000 203 0.000 406 0.054 0.000 633 8.732 605 8.731972 0.000 211 0.000 422 0.055 0.000 657 8.740 582 8.739 925 0.000 219 0.000 438 0.056 0.000 681 8.748 415 8.747 734 0.000 227 0.000 454 0.057 0.000 705 8.756 110 8.755 405 0.000 235 0.000 470 0.058 0.000 730 8.763 671 8.762 941 0.000 243 0.000 487 0.059 0.000 755 8.771 104 8.770 348 0.000 252 0.000 504 0.060 0.000 781 8.778 412 8.777 631 0.000 261 0.000 521 0.061 0.000 808 8.785 599 8.784 792 0.000 269 0.000 538 0.062 0.000 834 8.792 670 8.791836 0.000 278 0.000 556 0.063 0.000 861 8.799 628 8.798 767 0.000 287 0.000 574 0.064 0.000 889 8.806 476 8.805 588 0.000 296 0.000 592 0.065 0.000 917 8.813 219 8.812 302 0.000 306 0.000 611 0.066 0.000 945 8.819 859 8.818 914 0.000 315 0.000 630 0.067 0.000 974 8.826 400 8.825 426 0.000 325 0.000 649 0.068 0.001 003 8.832 844 8.831840 0.000 335 0.000 669 0.069 0.001 033 8.839 194 8.838 161 0.000 345 0.000 688 0.070 0.001063 8.845 453 8.844 390 0.000 355 0.000 709 0.071 0.001 094 8.851 623 8.850 529 0.000 365 0.000 729 0.072 0.001 125 8.857 708 8.856 583 0.000 375 0.000 750 0.073 0.001 156 8.863 709 8.862 552 0.000 386 0.000 770 0.074 0.001 188 8.869 628 8.868 440 0.000 396 0.000 792 0.075 0.001 220 8.875 468 8.874 248 0.000 407 0.000 813 0.076 0.001 253 8.881 232 8.879 979 0.000 418 0.000 835 0.077 0.001286 8.886 920 8.885 634 0.000 429 0.000 857 0.078 0.001320 8.892 535 8.891 215 0.000 440 0.000 880 0.079 0.001 354 8.898 079 8.896 725 0.000 452 0.000 902 0.080 0.001 388 8.903 553 8.902 165 0.000 463 0.000 925 0.081 0.001 423 8.908 960 8.907 537 0.000 475 0.000 948 0.082 0.001 458 8.914 300 8.912 842 0.000 487 0.000 972 0.083 0.001 494 8.919 577 8.918 082 0.000 499 0.000 996 0.084 0.001 530 8.924 790 8.923 260 0.000 511 0.001 020 0.085 0.001 567 8.929 942 8.928 375 0.000 523 0.001044 0.086 0.001604 8.935 034 8.933 430 0.000 535 0.001 069 0.087 0.001642 8.940 067 8.938 425 0.000 548 0.001 094 0.088 0.001 679 8.945 043 8.943 364 0.000 560 0.001 119 0.089 0.001 718 8.949 963 8.948 245 0.000 573 0.001 145 0.090 0.001 757 8.954 829 8.953 072 0.000 586 0.001 170 0.091 0.001 796 8.959 641 8.957 845 0.000 599 0.001 196 0.092 0.001835 8.964 400 8.962 565 0.000 612 0.001 223 0.093 0.001 875 8.969 109 8.967 233 0.000 626 0.001250 0.094 0.001 916 8.973 767 8.971 851 0.000 639 0.001 277 0.095 0.001 957 8.978 377 8.976 420 0.000 653 0.001 304 0.096 0.001998 8.982 938 8.980 940 0.000 667 0.001 331 0.097 0.002 040 8.987 453 8.985 413 0.000 681 0.001 359 0.098 0.002 082 8.991 921 8.989 839 0.000 695 0.001387 0.099 0.002 125 8.996 344 8.994 220 0.000 709 0.001 416 APPENDIX 287 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log cosh x log sinh x log tanh x . sinh x log — log, x tanhz 0.100 0.002 168 9.000 724 8.998 556 0.000 724 0.001444 0.101 0.002 211 9.005 059 9.002 848 0.000 738 0.001473 0.102 0.002 255 9.009 353 9.007 098 0.000 753 0.001502 0.103 0.002 300 9.013 605 9.011 305 0.000 768 0.001532 0.104 0.002 344 9.017 816 9.015 472 0.000 783 0.001 562 0.105 0.002 390 9.021987 9.019 597 0.000 798 0.001592 0.106 0.002 435 9.026 119 9.023 684 0.000 813 0.001622 0.107 0.002 481 9.030 212 9.027 731 0.000 828 0.001653 0.108 0.002 528 9.034 268 9.031 740 0.000 844 0.001 684 0.109 0.002 575 9.038 286 9.035 711 0.000 860 0.001 715 0.110 0.002 622 9.042 268 9.039 646 0.000 875 0.001 747 0.111 0.002 670 9.046 214 9.043 544 0.000 891 0.001 779 0.112 0.002 718 9.050126 9.047 407 0.000 908 0.001811 0.113 0.002 767 9.054 002 9.051235 0.000 924 0.001 843 0.114 0.002 816 9.057 845 9.055 029 0.000 940 0.001 876 0.115 0.002 865 9.061 655 9.058 789 0.000 957 0.001909 0.116 0.002 915 9.065 432 9.062 516 0.000 974 0.001 942 0.117 0.002 966 9.069 176 9.066 210 0.000 990 0.001975 0.118 0.003 017 9.072 889 9.069 873 0.001007 0.002 009 0.119 0.003 068 9.076 571 9.073 504 0.001025 0.002 043 0.120 0.003 119 9.080 223 9.077 104 0.001042 0.002 078 0.121 0.003 172 9.083 845 9.080 673 0.001059 0.002 112 0.122 0.003 224 9.087 437 9.084 213 0.001 077 0.002 147 0.123 0.003 277 9.091000 9.087 723 0.001 095 0.002 182 0.124 0.003 330 9.094 534 9.091 204 0.001 112 0.002 218 0.125 0.003 384 9.098 040 9.094 656 0.001 130 0.002 254 0.126 0.003 438 9.101519 9.098 081 0.001 149 0.002 290 0.127 0.003 493 9.104 971 9.101478 0.001 167 0.002 326 0.128 0.003 548 9.108 395 9.104 847 0.001 185 0.002 363 0.129 0.003 604 9.111794 9.108190 0.001204 0.002 400 0.130 0.003 659 9.115 166 9.111506 0.001223 0.002 437 0.131 0.003 716 9.118 513 9.114 797 0.001241 0.002 474 0.132 0.003 773 9.121834 9.118 062 0.001 260 0.002 512 0.133 0.003 830 9.125 131 9.121301 0.001280 •0.002 550 0.134 0.003 887 9.128 404 9.124 516 0.001299 0.002 589 0.135 0.003 946 9.131652 9.127 707 0.001 318 0.002 627 0.136 0.004 004 9.134 877 9.130 873 0.001338 0.002 666 0.137 0.004 063 9.138 078 9.134 015 0.001358 0.002 705 0.138 0.004 122 9.141257 9.137 134 0.001 378 0.002 745 0.139 0.004 182 9.144 412 9.140 230 0.001 398 0.002 784 0.140 0.004 242 9.147 546 9.143 304 0.001 418 0.002 824 0.141 0.004 303 9.150 657 9.146 354 0.001438 0.002 865 0.142 0.004 364 9.153 747 9.149 383 0.001 459 0.002 905 0.143 0.004 425 9.156 815 9.152 390 0.001 479 0.002 946 0.144 0.004 487 9.159 862 9.155 375 0.001500 0.002 987 0.145 0.004 550 9.162 889 9.158 339 0.001 521 0.003 029 0.146 0.004 612 9.165 895 9.161282 0.001542 0.003 071 0.147 0.004 676 9.168 880 9.164 205 0.001563 0.003 113 0.148 0.004 739 9.171846 9.167 107 0.001584 0.003 155 0.149 0.004 803 9.174 792 9.169 989 0.001606 0.003 197 2SS APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log cosh z log sinh x log tanh x sinh x log ,08 tanhx 0.150 0.004 868 9.177 719 9.172 851 0.001 627 0.003 240 0.151 0.004 932 9.180 626 9.175 694 0.001 649 0.003 283 0.152 0.004 998 9.183 515 9.178 517 0.001 671 0.003 327 0.153 0.005 063 9.186 385 9.181321 0.001 693 0.003 370 0.154 0.005 130 9.189 236 9.184 106 0.001 715 0.003 414 0.155 0.005 196 9.192 069 9.186 873 0.001 738 0.003 459 0.156 0.005 263 9.194 885 9.189 621 0.001760 0.003 503 0.157 0.005 331 9.197 682 9.192 352 0.001 783 0.003 548 0.158 0.005 398 9.200 463 9.195 064 0.001 805 0.003 593 0.159 0.005 467 9.203 225 9.197 759 0.001 828 0.003 638 0.160 0.005 535 9.205 971 9.200 436 0.001851 0.003 684 0.161 0.005 605 9.208 700 9.203 096 0.001 875 0.003 730 0.162 0.005 674 9.211413 9.205 739 0.001898 0.003 776 0.163 0.005 744 9.214 109 9.208 365 0.001 921 0.003 823 0.164 0.005 814 9.216 789 9.210 975 0.001945 0.003 869 0.165 0.005 885 9.219 453 9.213 568 0.001 969 0.003 916 0.166 0.005 956 9.222 101 9.216 144 0.001 993 0.003 964 0.167 0.006 028 9.224 733 9.218 705 0.002 017 0.004 011 0.168 0.006 100 9.227 350 9.221 250 0.002 041 0.004 059 0.169 0.006 173 9.229 952 9.223 779 0.002 065 0.004 107 0.170 0.006 246 9.232 539 9.226 293 0.002 090 0.004156 0.171 0.006 319 9.235 111 9.228 792 0.002 114 0.004 204 0.172 0.006 393 9.237 668 9.231 275 0.002 139 0.004 253 0.173 0.006 467 9.240 210 9.233 743 0.002 164 0.004 303 0.174 0.006 541 9.242 738 9.236 197 0.002 189 0.004 352 0.175 0.006 616 9.245 253 9.238 636 0.002 214 0.004 402 0.176 0.006 692 9.247 752 9.241 061 0.002 240 0.004 452 0.177 0.006 768 9.250 239 9.243 471 0.002 265 0.004 502 0.178 0.006 844 9.252 711 9.245 867 0.002 291 0.004 553 0.179 0.006 921 9.255 170 9.248 249 0.002 317 0.004 604 0.180 0.006 998 9.257 615 9.250 617 0.002 343 0.004 655 0.181 0.007 075 9.260 047 9.252 972 0.002 369 0.004 707 0.182 0.007 153 9.262 466 9.255 313 0.002 395 0.004 758 0.183 0.007 232 9.264 872 9.257 641 0.002 421 0.004 810 0.184 0.007 311 9.267 266 9.259 955 0.002 448 0.004 863 0.185 0.007 390 9.269 646 9.262 256 0.002 474 0.004 915 0.186 0.007 470 9.272 014 9.264 545 0.002 501 0.004 968 0.187 0.007 550 9.274 370 9.266 820 0.002 528 0.005 021 0.188 0.007 630 9.276 713 9.269 083 0.002 555 0.005 075 0.189 0.007 711 9.279 044 9.271333 0.002 583 0.005 128 0.190 0.007 792 9.281 363 9.273 571 0.002 610 0.005 182 0.191 0.007 874 9.283 671 9.275 797 0.002 637 0.005 237 0.192 0.007 956 9.285 966 9.278 010 0.002 665 0.005 291 0.193 0.008 039 9.288 250 9.280 211 0.002 693 0.005 346 0.194 0.008 122 9.290 523 9.282 401 0.002 721 0.005 401 0.195 0.008 205 9.292 783 9.284 578 0.002 749 0.005 456 0.196 0.008 289 9.295 033 9.286 744 0.002 777 0.005 512 0.197 0.008 373 9.297 272 9.288 898 0.002 805 0.005 568 0.198 0.008 458 9.299 499 9.291041 0.002 834 0.005 624 0.199 0.008 543 9.301 716 9.293 173 0.002 863 0.005 680 APPENDIX 289 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log ainh x log tanh x , sinh x log X X X tanh i 0.200 0.008 629 9.303 921 9.295 293 0.002 891 0.005 737 0.201 0.008 715 9.306 116 9.297 402 0.002 920 0.005 794 0.202 0.008 801 9.308 301 9.299 500 0.002 949 0.005 851 0.203 0.008 888 9.310 475 9.301587 0.002 979 0.005 909 0.204 0.008 975 9.312 638 9.303 663 0.003 008 0.005 967 0.205 0.009 062 9.314 791 9.305 729 0.003 038 0.006 025 0.206 0.009 150 9.316 935 9.307 784 0.003 067 0.006 083 0.207 0.009 239 9.319 067 9.309 829 0.003 097 0.006 142 0.208 0.009 328 9.321 190 9.311 863 0.003 127 0.006 201 0.209 0.009 417 9.323 303 9.313 886 0.003 157 0.006 260 0.210 0.009 507 9.325 407 9.315 900 0.003 187 0.006 319 0.211 0.009 597 9.327 500 9.317 904 0.003 218 0.006 379 0.212 0.009 687 9.329 584 9.319 897 0.003 248 0.006 439 0.213 0.009 778 9.331 659 9.321 880 0.003 279 0.006 499 0.214 0.009 869 9.333 724 9.323 854 0.003 310 0.006 560 0.215 0.009 961 9.335 779 9.325 818 0.003 341 0.006 621 0.216 0.010 053 9.337 826 9.327 772 0.003 372 0.006 682 0.217 0.010 146 9.339 863 9.329 717 0.003 403 0.006 743 0.218 0.010 239 9.341 891 9.331652 0.003 434 0.006 805 0.219 0.010 332 9.343 910 9.333 578 0.003 466 0.006 866 0.220 0.010 426 9.345 920 9.335 494 0.003 498 0.006 929 0.221 0.010 520 9.347 922 9.337 401 0.003 529 0.006 991 0.222 0.010 615 9.349 914 9.339 299 0.003 561 0.007 054 0.223 0.010 710 9.351898 9.341 188 0.003 594 0.007 117 0.224 0.010 806 9.353 874 9.343 068 0.003 626 0.007 180 0.225 0.010 902 9.355 841 9.344 939 0.003 658 0.007 243 0.226 0.010 998 9.357 799 9.346 801 0.003 691 0.007 307 0.227 0.011095 9.359 749 9.348 655 0.003 723 0.007 371 0.228 0.011 192 9.361 691 9.350 499 0.003 756 0.007 435 0.229 0.011 289 9.363 625 9.352 335 0.003 789 0.007 500 0.230 0.011387 9.365 550 9.354 163 0.003 822 0.007 565 0.231 0.011 486 9.367 468 9.355 982 0.003 856 0.007 630 0.232 0.011 584 9.369 377 9.357 793 0.003 889 0.007 695 0.233 0.011 684 9.371 278 9.359 595 0.003 922 0.007 761 0.234 0.011783 9.373 172 9.361 389 0.003 956 0.007 827 0.235 0.011883 9.375 058 9.363 175 0.003 990 0.007 893 0.236 0.011984 9.376 936 9.364 952 0.004 024 0.007 960 0.237 0.012 084 9.378 806 9.366 722 0.004 058 0.008 026 0.238 0.012 186 9.380 669 9.368 484 0.004 092 0.008 093 0.239 0.012 287 9.382 525 ,9.370 237 0.004 127 0.008 161 0.240 0.012 389 9.384 372 9.371983 0.004 161 0.008 228 0.241 0.012 492 9.386 213 9.373 721 0.004 196 0.008 296 0.242 0.012 595 9.388 046 9.375 451 0.004 231 ■ 0.008 364 0.243 0.012 698 9.389 872 9.377 174 0.004 266 0.008 432 0.244 0.012 802 9.391 691 9.378 889 0.004 301 0.008 501 0.245 0.012 906 9.393 502 9.380 596 0.004 336 0.008 570 0.246 0.013 010 9.395 307 9.382 296 0.004 371 0.008 639 0.247 0.013 115 9.397 104 9.383 989 0.004 407 0.008 708 0.248 0.013 221 9.398 894 9.385 674 0.004 443 0.008 778 0.249 0.013 326 9.400 678 9.387 351 0.004 479 0.008 848 200 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log ooeh x log sinh x log tanh z . sinh z log log: — c - tanh i 0.250 0.013 433 9.402 455 9.389 022 0.004 515 0.008 918 0.251 0.013 539 9.404 224 9.390 685 0.004 551 0.008 989 0.252 0.013 646 9.405 987 9.392 341 0.004 587 0.009 059 0.253 0.013 754 9.407 744 9.393 990 0.004 623 0.009 130 0.254 0.013 861 9.409 494 9.395 632 0.004 660 0.009 202 0.255 0.013 970 9.411 237 9.397 267 0.004 697 0.009 273 0.256 0.014 078 9.412 973 9.398 895 0.004 733 0.009 345 0.257 0.014 187 9.414 703 9.400 516 0.004 770 0.009 417 0.258 0.014 297 9.416 427 9.402 130 0.004 807 0.009 489 0.259 0.014 406 9.418 144 9.403 738 0.004 845 0.009 562 0.260 0.014 517 9.419 855 9.405 339 0.004 882 0.009 635 0.261 0.014 627 9.421560 9.406 933 0.004 920 0.009 708 0.262 0.014 738 9.423 259 9.408 520 0.004 957 0.009 781 0.263 0.014 850 9.424 951 9.410 101 0.004 995 0.009 855 0.264 0.014 962 9.426 637 9.411 675 0.005 033 0.009 929 0.265 0.015 074 9.428 317 9.413 243 0.005 071 0.010 003 0.266 0.015 187 9.429 991 9.414 804 0.005 109 0.010 077 0.267 0.015 300 9.431 659 9.416 359 0.005 148 0.010 152 0.268 0.015 413 9.433 321 9.417 908 0.005 186 0.010 227 0.269 0.015 527 9.434 977 9.419 450 0.005 225 0.010 302 0.270 0.015 641 9.436 628 9.420 986 0.005 264 0.010 377 0.271 0.015 756 9.438 272 9.422 516 0.005 303 0.010 453 0.272 0.015 871 9.439 911 9.424 040 0.005 342 0.010 529 0.273 0.015 987 9.441544 9.425 557 0.005 381 0.010 605 0.274 0.016 103 9.443 171 9.427 069 0.005 421 0.010 682 0.275 0.016 219 9.444 793 9.428 574 0.005 460 0.010 759 0.276 0.016 336 9.446 409 9.430 073 0.005 500 0.010 836 0.277 0.016 453 9.448 019 9.431567 0.005 540 0.010 913 0.278 0.016 570 9.449 624 9.433 054 0.005 580 0.010 991 0.279 0.016 688 9.451224 9.434 536 0.005 620 0.011068 0.280 0.016 806 9.452 818 9.436 012 0.005 660 0.011 146 0.281 0.016 925 9.454 407 9.437 482 0.005 700 0.011225 0.282 0.017 044 9.455 990 9.438 946 0.005 741 0.011 303 0.283 0.017 164 9.457 568 9.440 404 0.005 782 0.011382 0.284 0.017 284 9.459 141 9.441 857 0.005 822 0.011461 0.285 0.017 404 9.460 708 9.443 304 0.005 863 0.011541 0.286 0.017 525 9.462 271 9.444 746 0.005 905 0.011 620 0.287 0.017 646 9.463 828 9.446 182 0.005 946 0.011 700 0.288 0.017 767 9.465 380 9.447 612 0.005 987 0.011 780 0.289 0.017 889 9.466 927 9.449 037 0.006 029 0.011 861 0.290 0.018 012 9.468 468 9.450 457 0.006 070 0.011 941 0.291 0.018 134 9.470 005 9.451 871 0.006 112 0.012 022 0.292 0.018 258 9.471 537 9.453 279 0.006 154 0.012 103 0.293 0.018 381 9.473 064 9.454 683 0.006 196 0.012 185 0.294 0.018 505 9.474 586 9.456 081 0.006 239 0.012 267 0.295 0.018 629- 9.476 103 9.457 474 0.006 281 0.012 348 0.296 0.018 754 9.477 615 9.458 861 0.006 323 0.012 431 0.297 0.018 879 9.479 123 9.460 243 0.006 366 0.012 513 0.298 0.019 005 9.480 625 9.461 620 0.006 409 0.012 596 . 0.299 0.019 131 9.482 123 9.462 992 0.006 452 0.012 679 APPENDIX 291 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log cosh z log sinh x log tanh x . sinh x log lo 8; — j— tanh x 0.300 0.019 257 9.483 616 9.464 359 0.006 495 0.012 762 0.301 0.019 384 9.485 105 9.465 721 0.006 538 0.012 845 0.302 0.019 511 9.486 589 9.467 078 0.006 582 0.012 929 0.303 0.019 638 9.488 068 9.468 429 0.006 625 0.013 013 0.304 0.019 766 9.489 542 9.469 776 0.006 669 0.013 097 0.305 0.019 894 9.491 012 9.471 118 0.006 713 0.013 182 0.306 0.020 023 9.492 478 9.472 455 0.006 757 0.013 267 0.307 0.020 152 9.493 939 9.473 787 0.006 801 0.013 352 0.308 0.020 282 9.495 396 9.475 114 0.006 845 0.013 437 0.309 0.020 412 9.496 848 9.476 436 0.006 889 0.013 522 0.310 0.020 542 9.498 295 9.477 754 0.006 934 0.013 608 0.311 0.020 673 9.499 739 9.479 066 0.006 978 0.013 694 0.312 0.020 804 9.501 178 9.480 374 0.007 023 0.013 780 0.313 0.020 935 9.502 613 9.481 677 0.007 068 0.013 867 0.314 0.021 067 9.504 043 9.482 976 0.007 113 0.013 954 0.315 0.021 199 9.505 469 9.484 270 0.007 159 0.014 041 0.316 0.021 332 9.506 891 9.485 559 0.007 204 0.014 128 0.317 0.021 465 9.508 309 9.486 844 0.007 249 0.014 216 0.318 0.021 598 9.509 722 9.488 124 0.007 295 0.014 303 0.319 0.021 732 9.511 132 9.489 399 0.007 341 0.014 391 0.320 0.021 866 9.512 537 9.490 670 0.007 387 0.014 480 0.321 0.022 001 9.513 938 9.491 937 0.007 433 0.014 568 0.322 0.022 136 9.515 335 9.493 199 0.007 479 0.014 657 0.323 0.022 271 9.516 728 9.494 457 0.007 525 0.014 746 0.324 0.022 407 9.518117 9.495 710 0.007 572 0.014 835 0.325 0.022 543 9.519 502 9.496 959 0.007 619 0.014 925 0.326 0.022 680 9.520 883 9.498 203 0.007 665 0.015 015 0.327 0.022 817 9.522 260 9.499 443 0.007 712 0.015 105 0.328 0.022 954 9.523 633 9.500 679 0.007 759 0.015 195 0.329 0.023 092 9.525 003 9.501911 0.007 807 0.015 285 0.330 0.023 230 9.526 368 9.503 138 0.007 854 0.015 376 0.331 0.023 369 9.527 730 9.504 361 0.007 902 0.015 467 0.332 0.023 508 9.529 087 9.505 580 0.007 949 0.015 558 0.333 0.023 647 9.530 441 9.506 794 0.007 997 0.015 650 0.334 0.023 787 9.531 791 9.508 005 0.008 045 0.015 742 0.335 0.023 927 9.533 138 9.509 211 0.008 093 0.015 834 0.336 0.024 067 9.534 480 9.510 413 0.008 141 0.015 926 0.337 0.024 208 9.535 819 9.511 611 0.008 190 0.016 019 0.338 0.024 349 9.537 155 9.512 805 0.008 238 0.016 111 0.339 0.024 491 9.538 486 9.513 995 0.008 287 0.016 204 0.340 0.024 633 9.539 814 9.515 181 0.008 335 0.016 298 0.341 0.024 775 9.541 139 9.516 363 0.008 384 0.016 391 0.342 0.024 918 9.542 459 9.517 541 0.008 433 0.016 485 0.343 0.025 061 9.543 777 9.518 715 0.008 483 0.016 579 0.344 0.025 205 9.545 090 9.519 885 0.008 532 0.016 673 0.345 0.025 349 9.546 400 9.521 052 0.008 581 0.016 768 0.346 0.025 493 9.547 707 9.522 214 0.008 631 0.016 862 0.347 0.025 638 9.549 010 9.523 372 0.008 681 0.016 957 0.348 0.025 783 9.550 310 9.524 527 0.008 731 0.017 052 0.349 0.025 929 9.551 606 9.525 678 0.008 781 0.017 148 202 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log cosh z log sinh z log tan h z , sinh z log log; T— 0.350 0.026 075 9.552 899 9.526 824 0.008 831 0.017 244 0.351 0.026 221 9.554 188 9.527 968 0.008 881 0.017 340 0.352 026 367 9.555 474 9.529 107 0.008 932 0.017 436 0.353 0.026 515 9.556 757 9.530 242 0.008 982 0.017 532 0.354 0.026 662 9.558 036 9.531 374 0.009 033 0.017 629 0.355 0.026 810 9.559 312 9.532 503 0.009 084 0.017 726 0.356 0.026 958 9.560 585 9.533 627 0.009 135 0.017 823 0.357 0.027 107 9.561854 9.534 748 0.009 186 0.017 920 0.358 0.027 256 9.563 121 9.535 865 0.009 238 0.018 018 0.359 0.027 405 9.564 383 9.536 979 0.009 289 0.018 116 0.360 0.027 555 9.565 643 9.538 088 0.009 341 0.018 214 0.361 0.027 705 9.566 900 9.539 195 0.009 392 0.018 312 0.362 0.027 855 9.568 153 9.540 298 0.009 444 0.018 411 0.363 0.028 006 9.569 403 9.541 397 0.009 496 0.018 510 0.364 0.028 157 9.570 650 9.542 492 0.009 548 0.018 609 0.365 0.028 309 9.571894 9.543 585 0.009 601 0.018 708 0.366 0.028 461 9.573 134 9.544 673 0.009 653 0.018 808 0.367 0.028 613 9.574 372 9.545 758 0.009 706 0.018 908 0.368 0.028 766 9.575 606 9.546 840 0.009 758 0.019 008 0.369 0.028 919 9.576 838 9.547 918 0.009 811 0.019 108 0.370 0.029 073 9.578 066 9.548 993 0.009 864 0.019 209 0.371 0.029 227 9.579 291 9.550 065 0.009 917 019 309 0.372 0.029 381 9.580 514 9.551 133 0.009 971 0.019 410 0.373 0.029 536 9.581 733 9.552 197 0.010 024 0.019 512 0.374 0.029 691 9.582 949 9.553 258 0.010 078 0.019 613 0.375 0.029 846 9.584 163 9.554 316 0.010 131 0.019 715 0.376 0.030 002 9.585 373 9.555 371 0.010 185 0.019 817 0.377 0.030 158 9.586 581 9.556 422 0.010 239 0.019 919 0.378 0.030 315 9.587 785 9.557 470 0.010 293 0.020 022 0.379 0.030 472 9.588 987 9.558 515 0.010 348 0.020 124 0.380 0.030 629 9.590 186 9.559 556 0.010 402 0.020 227 0.381 0.030 787 9.591382 9.560 595 0.010 457 0.020 330 0.382 0.030 945 9.592 575 9.561 630 0.010 511 0.020 434 0.383 0.031 104 9.593 765 9.562 662 0.010 566 0.020 537 0.384 0.031 262 9.594 952 9.563 690 0.010 621 0.020 641 0.385 0.031422 9.596 137 9.564 716 0.010 676 0.020 745 0.386 0.031581 9.597 319 9.565 738 0.010 732 0.020 850 0.387 0.031741 9.598 498 9.566 757 0.010 787 0.020 954 0.388 0.031 901 9.599 674 9.567 773 0.010 843 0.021059 0.389 0.032 062 9.600 848 9.568 786 0.010 898 0.021 164 0.390 0.032 223 9.602 019 9.569 795 0.010 954 0.021 269 0.391 0.032 385 9.603 187 9.570 802 0.011010 0.021 375 0.392 0.032 547 9.604 352 9.571 806 0.011066 0.021480 0.393 0.032 709 9.605 515 9.572 806 0.011 122 0.021586 0.394 0.032 871 9.606 675 9.573 804 0.011 179 0.021 693 0.395 0.033 034 9.607 832 9.574 798 0.011 235 0.021 799 0.396 0.033 198 9.608 987 9.575 789 0.011292 0.021906 0.397 0.033 361 9.610 139 9.576 778 0.011 349 0.022 013 0.398 0.033 525 9.611 289 9.577 763 0.011406 0.022 120 0.399 0.033 690 9.612 436 9.578 746 0.011463 0.022 227 APPENDIX 293 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log cosh x log sinh x log tanh x . sinh x log X log. ' tanh x 0.400 0.033 855 9.613 580 9.579 725 0.011 520 0.022 335 0.401 0.034 020 9.614 722 9.580 702 0.011 577 0.022 443 0.402 0.034 186 9.615 861 9.581 675 0.011 635 0.022 551 0.403 0.034 351 9.616 998 9.582 646 0.011 693 0.022 659 0.404 0.034 518 9.618 132 9.583 614 0.011 750 0.022 767 0.405 0.034 684 9.619 263 9.584 579 0.011 808 0.022 876 0.406 0.034 852 9.620 392 9.585 541 0.011 866 0.022 985 0.407 0.035 019 9.621519 9.586 500 0.011925 0.023 094 0.408 0.035 187 9.622 643 9.587 456 0.011 983 0.023 204 0.409 0.035 355 9.623 765 9.588 410 0.012 041 0.023 313 0.410 0.035 523 9.624 884 9.589 360 0.012 100 0.023 423 0.411 0.035 692 9.626 001 9.590 308 0.012 159 0.023 534 0.412 0.035 862 9.627 115 9.591 253 0.012 218 0.023 644 0.413 0.036 031 9.628 227 9.592 196 0.012 277 0.023 754 0.414 0.036 201 9.629 336 9.593 135 0.012 336 0.023 865 0.415 0.036 372 9.630 443 9.594 072 0.012 395 0.023 976 0.416 0.036 542 9.631548 9.595 006 0.012 455 0.024 088 0.417 0.036 713 9.632 650 9.595 937 0.012 514 0.024 199 0.418 0.036 885 9.633 750 9.596 866 0.012 574 0.024 311 0.419 0.037 057 9.634 848 9.597 791 0.012 634 0.024 423 0.420 0.037 229 9.635 943 9.598 714 0.012 694 0.024 535 0.421 0.037 401 9.637 036 9.599 635 0.012 754 0.024 647 0.422 0.037 574 9.638 127 9.600 553 0.012 814 0.024 760 0.423 0.037 748 9.639 215 9.601 468 0.012 875 0.024 873 0.424 0.037 921 9.640 301 9.602 380 0.012 936 0.024 986 0.425 0.038 095 9.641 385 9.603 290 0.012 996 0.025 099 0.426 0.038 270 9.642 467 9.604 197 0.013 057 0.025 213 0.427 0.038 445 9.643 546 9.605 101 0.013 118 0.025 326 0.428 0.038 620 9.644 623 9.606 003 0.013 179 0.025 440 0.429 0.038 795 9.645 698 9.606 903 0.013 241 0.025 555 0.430 0.038 971 9.646 770 9.607 799 0.013 302 0.025 669 0.431 0.039 147 9.647 841 9.608 694 0.013 364 0.025 784 0.432 0.039 324 9.648 909 9.609 585 0.013 425 0.025 899 0.433 0.039 501 9.649 975 9.610 474 0.013 487 0.026 014 0.434 0.039 678 9.651039 9.611 361 0.013 549 0.026 129 0.435 0.039 856 9.652 100 9.612 245 0.013 611 0.026 244 0.436 0.040 034 9.653 160 9.613 126 0.013 673 0.026 360 0.437 0.040 212 9.654 217 9.614 005 0.013 736 0.026 476 0.438 0.040 391 9.655 273 9.614 882 0.013 798 0.026 592 0.439 0.040 570 9.656 326 9.615 756 0.013 861 0.026 709 0.440 0.040 749 9.657 377 9.616 627 0.013 924 0.026 825 0.441 0.040 929 9.658 425 9.617 496 0.013 987 0.026 942 0.442 0.041 109 9.659 472 9.618 363 0.014 050 0.027 059 0.443 0.041 290 9.660 517 9.619 227 0.014 113 0.027 177 0.444 0.041 471 9.661 560 9.620 089 0.014 177 0.027 294 0.445 0.041 652 9.662 600 9.620 948 0.014 240 0.027 412 0.446 0.041 834 9.663 639 9.621805 0.014 304 0.027 530 0.447 0.042 016 9.664 675 9.622 659 0.014 368 0.027 648 0.448 0.042 198 9.665 709 9.623 511 0.014 431 0.027 767 0.449 0.042 381 9.666 742 9.624 361 0.014 496 0.027 885 294 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) X log cosh z log Hlllh 1 log tanh x , .-mil X log * tanh x 0.450 0.042 564 9.667 772 9.625 209 0.014 560 0.028 004 0.451 0.042 747 9.668 801 9.626 053 0.014 624 0.028 123 0.452 0.042 931 9.669 827 9.626 896 0.014 689 0.028 242 0.453 0.043 115 9.670 851 9.627 736 0.014 753 0.028 362 0.454 0.043 300 9.671 874 9.628 574 0.014 818 0.028 482 0.455 0.043 484 9.672 894 9.629 410 0.014 883 0.028 601 0.456 0.043 670 9.673 913 9.630 243 0.014 948 0.028 722 0.457 0.043 855 9.674 929 9.631074 0.015 013 0.028 842 0.458 0.044 041 9.675 944 9.631 903 0.015 078 0.028 963 0.459 0.044 227 9.676 957 9.632 729 0.015 144 0.029 083 0.460 0.044 414 9.677 967 9.633 553 0.015 210 0.029 204 0.461 0.044 601 9.678 976 9.634 375 0.015 275 0.029 326 0.462 0.044 788 9.679 983 9.635 195 0.015 341 0.029 447 0.463 0.044 976 9.680 988 9.636 012 0.015 407 0.029 569 0.464 0.045 164 9.681991 9.636 827 0.015 473 0.029 690 0.465 0.045 352 9.682 993 9.637 640 0.015 540 0.029 813 0.466 0.045 541 9.683 992 9.638 451 0.015 606 0.029 935 0.467 0.045 730 9.684 989 9.639 260 0.015 673 0.030 057 0.468 0.045 919 9.685 985 9.640 066 0.015 739 0.030 180 0.469 0.046 109 9.686 979 9.640 870 0.015 806 0.030 303 0.470 0.046 299 9.687 971 9.641672 0.015 873 0.030 426 0.471 0.046 490 9.688 961 9.642 471 0.015 940 0.030 550 0.472 0.046 681 9.689 950 9.643 269 0.016 008 0.030 673 0.473 0.046 872 9.690 936 9.644 064 0.016 075 0.030 797 0.474 0.047 063 9.691 921 9.644 857 0.016 143 0.030 921 0.475 0.047 255 9.692 904 9.645 649 0.016 210 0.031045 0.476 0.047 447 9.693 885 9.646 437 0.016 278 0.031 169 0.477 0.047 640 9.694 864 9.647 224 0.016 346 0.031294 0.478 0.047 833 9.695 842 9.648 009 0.016 414 0.031419 0.479 0.048 026 9.696 818 9.648 792 0.016 482 0.031544 0.480 0.048 220 9.697 792 9.649 572 0.016 551 0.031669 0.481 0.048 414 9.698 764 9.650 350 0.016 619 0.031795 0.482 0.048 608 9.699 735 9.651 127 0.016 688 0.031920 0.483 0.048 803 9.700 704 9.651901 0.016 757 0.032 046 0.484 0.048 998 9.701671 9.652 673 0.016 826 0.032 172 0.485 0.049 193 9.702 636 9.653 443 0.016 895 0.032 299 0.486 0.049 389 9.703 600 9.654 211 0.016 964 0.032 425 0.487 0.049 585 9.704 562 9.654 977 0.017 033 0.032 552 0.488 0.049 781 9.705 522 9.655 741 0.017 103 0.032 679 0.489 0.049 978 9.706 481 9.656 503 0.017 172 0.032 806 0.490 0.050 175 9.707 438 9.657 263 0.017 242 0.032 933 0.491 0.050 373 9.708 393 9.658 021 0.017 312 0.033 061 0.492 0.050 570 9.709 347 9.658 777 0.017 382 0.033 189 0.493 0.050 769 9.710 299 9.659 530 0.017 452 0.033 316 0.494 0.050 967 9.711 249 9.660 282 0.017 522 0.033 445 0.495 0.051 166 9.712 198 9.661 032 0.017 593 0.033 573 0.496 0.051 365 9.713 145 9.661 780 0.017 663 0.033 702 0.497 0.051 565 9.714 091 9.662 526 0.017 734 0.033 830 0.498 0.051 764 9.715 034 9.663 270 0.017 805 0.033 959 0.499 0.051 965 9.715 977 9.664 012 0.017 876 0.034 089 APPENDIX 295 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log si li li X log tanh x 0.500 0.501 0.502 0.503 0.504 0.505 0.506 0.507 0.508 0.509 0.510 0.511 0.512 0.513 0.514 0.515 0.516 0.517 0.518 0.519 0.520 0.521 0.522 0.523 0.524 0.525 0.526 0.527 0.528 0.529 0.530 0.531 0.532 0.533 0.534 0.535 0.536 0.537 0.538 0.539 0.540 0.541 0.542 0.543 0.544 0.545 0.546 0.547 0.548 0.549 0.052 165 0.052 366 0052 567 0.052 769 0.052 971 0.053 173 0.053 375 0.053 578 0.053 782 0.053 985 0.054 189 0.054 393 0.054 598 0.054 803 0.055 008 .055 214 0.055 420 0.055 626 0.055 833 0.056 040 0.056 247 0.056 454 0.056 662 0.056 871 0.057 079 0.057 288 0.057 498 0.057 707 0.057 917 0.058 128 0.058338 0.058 549 0.058 760 0.058 972 0.059 184 0.059 396 0.059 609 0.059 822 0.060 "035 0.060 249 0.060 463 0.060 677 0.060 892 0.061 106 0.061322 0.061 537 0.061 753 0.061 969 0.062 186 0.062 403 201 201 202 202 202 202 203 204 203 204 204 205 205 205 206 206 206 207 207 207 207 208 209 208 209 210 209 210 211 210 211 211 212 212 212 213 213 213 214 214 214 215 214 216 215 216 216 217 217 217 9.716 917 9.717 856 9.718 794 9.719 729 9.720 664 9.721 596 9.722 527 9.723 457 9.724 385 9.725 311 9.726 236 9.727 160 9.728 081 9.729 002 9.729 921 9.730 838 9.731 754 9.732 668 9.733 581 9.734 492 9.735 402 9.736 311 9.737 217 9.738 123 9.739 027 9.739 930 9.740 831 9.741 730 9.742 629 9.743 526 9.744 421 9.745 315 9.746 208 9.747 099 9.747 989 9.748 877 9.749 764 9.750 650 9.751 534 9.752 417 9.753 299 9.754 179 9.755 058 9.755 936 9.756 812 9.757 687 9.758 561 9.759 433 9.760 304 9.761 173 939 938 935 935 932 931 930 928 926 925 924 921 921 919 917 916 914 913 911 910 909 906 906 904 903 901 899 899 897 895 894 893 891 890 888 887 886 884 883 882 880 879 878 876 875 874 872 871 869 869 9.664 752 9.665 490 9.666 226 9.666 961 9.667 693 9.668 423 9.669 152 9.669 879 9.670 603 9.671326 9.672 047 9.672 766 9.673 484 9.674 199 9.674 913 9.675 624 9.676 334 9.677 042 9.677 748 9.678 453 9.679 155 9.679 856 9.680 555 9.681 252 9.681 948 9.682 641 9.683 333 9.684 023 9.684 711 9.685 398 9.686 083 9.686 766 9.687 447 9.688 127 9.688 805 9.689 481 9.690 155 9.690 828 9.691 499 9.692 169 9.692 836 9.693 502 9.694 167 9.694 829 9.695 490 9.696 150 9.696 807 9.697 463 9.698 118 9.698 770 738 736 735 732 730 729 727 724 723 721 719 718 715 714 711 710 708 706 705 702 701 699 697 696 693 692 690 688 687 685 683 681 680 678 676 674 673 671 670 667 666 665 662 661 660 657 656 655 652 652 290 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log coah z J log sinh x log tanh x 0.550 0.551 0.552 0.553 0.554 0.555 0.556 0.557 0.558 0.559 0.560 0.561 0.562 0.563 0.564 0.565 0.566 0.567 0.568 0.569 0.570 0.571 0.572 0.573 0.574 0.575 0.576 0.577 0.578 0.579 0.580 0.581 0.582 0.583 0.584 0.585 0.586 0.587 0.588 0.589 0.590 0.591 0.592 0.593 0.594 0.595 0.596 0.597 0.598 0.599 0.062 620 0.062 838 0.063 056 0.063 274 0.063 492 0.063 711 0.063 930 0.064 150 0.064 369 0.064 590 0.064 810 0.065 031 0.065 252 0.065 473 0.065 695 0.065 917 0.066 140 0.066 362 0.066 585 0.066 809 0.067 032 0.067 256 0.067 481 0.067 705 0.067 930 0.068 155 0.068 381 0.068 607 0.068 833 0.069 059 0.069 286 0.069 513 0.069 741 0.069 969 0.070 197 0.070 425 0.070 654 0.070 883 0.071 112 0.071 342 0.071 572 0.071 802 0.072 033 0.072 264 0.072 495 0.072 726 0.072 958 0.073 190 0.073 423 0.073 656 218 218 218 218 219 219 220 219 221 220 221 221 221 222 222 223 222 223 224 223 224 225 224 225 225 226 226 226 226 227 227 228 228 228 228 229 229 229 230 230 230 231 231 231 231 232 232 233 233 233 9.762 042 9.762 909 9.763 775 9.764 639 9.765 502 9.766 364 9.767 225 9.768 084 9.768 942 9.769 799 9.770 654 9.771 509 9.772 362 9.773 214 9.774 064 9.774 914 9.775 762 9.776 609 9.777 455 9.778 299 9.779 142 9.779 984 9.780 825 9.781 665 9.782 504 9.783 341 9.784 177 9.785 012 9.785 846 9.786 679 9.787 510 9.788 340 9.789 170 9.789 998 9.790 825 9.791 650 9.792 475 9.793 298 9.794 121 9.794 942 9.795 762 9.796 581 9.797 399 9.798 216 9.799 032 9.799 846 9.800 660 9.801 472 9.802 284 9.803 094 867 866 864 863 862 861 859 858 857 855 855 853 852 850 850 848 847 846 844 843 842 841 840 839 837 836 835 834 833 831 830 830 828 827 825 825 823 823 821 820 819 818 817 816 814 814 812 812 810 809 9.699 422 9.700 071 9.700 719 9.701 365 9.702 010 9.702 653 9.703 294 9.703 934 9.704 573 9.705 209 9.705 844 9.706 478 9.707 110 9.707 740 9.708 369 9.708 997 9.709 622 9.710 247 9.710 869 9.711 490 9.712 110 9.712 728 9.713 345 9.713 960 9.714 574 9.715 186 9.715 796 9.716 405 9.717 013 9.717 619 9.718 224 9.718 827 9.719 429 9.720 029 9.720 628 9.721 225 9.721 821 9.722 416 9.723 009 9.723 600 9.724 190 9.724 779 9.725 366 9.725 952 9.726 537 9.727 120 9.727 702 9.728 282 9.728 861 9.729 438 649 648 646 645 643 641 640 639 636 635 634 632 630 629 628 625 625 622 621 620 618 617 615 614 612 610 609 608 606 605 603 602 600 599 597 596 595 593 591 590 589 587 586 585 583 582 580 579 577 576 APPENDIX 297 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh x log tanh x 0.600 0.601 0.602 0.603 0.604 0.605 0.606 0.607 0.608 0.609 0.610 0.611 0.612 0.613 0.614 0.615 0.616 0.617 0.618 0.619 0.620 0.621 0.622 0.623 0.624 0.625 0.626 0.627 0.628 0.629 0.630 0.631 0.632 0.633 0.634 0.635 0.636 0.637 0.638 0.639 0.640 0.641 0.642 0.643 0.644 0.645 0.646 0.647 0.648 0.649 0.073 889 0.074 122 0.074 356 0.074 590 0.074 824 0.075 059 0.075 294 0.075 529 0.075 765 0.076 000 0.076 237 0.076 473 0.076 710 0.076 947 0.077 184 0.077 422 0.077 660 0.077 898 0.078 137 0.078 376 0.078 615 0.078 854 0.079 094 0.079 334 0.079 575 0.079 815 0.080 056 0.080 298 0.080 539 0.080 781 0.081024 0.081 266 0.081509 0.081 752 0.081 995 0.082 239 0.082 483 0.082 727 0.082 972 0.083 217 0.083 462 0.083 707 0.083 953 0.084 199 0.084 446 0.084 692 0.084 939 0.085 187 0.085 434 0.085 682 233 234 234 234 235 235 235 236 235 237 236 237 237 237 238 238 238 239 239 239 239 240 240 241 240 241 242 241 242 243 242 243 243 243 244 244 244 245 245 245 245 246 246 247 246 247 248 247 248 248 9.803 903 9.804 711 9.805 518 9.806 324 9.807 129 9.807 933 9.808 736 9.809 538 9.810 339 9.811 138 9.811 937 9.812 735 9.813 531 9.814 327 9.815 121 9.815 915 9.816 707 9.817 499 9.818 289 9.819 079 9.819 867 9.820 655 9.821441 9.822 227 9.823 012 9.823 795 9.824 578 9.825 359 9.826 140 9.826 920 9.827 698 9.828 476 9.829 253 9.830 029 9.830 804 9.831 578 9.832 351 9.833 123 9.833 894 9.834 664 9.835 433 9.836 202 9.836 969 9.837 735 9.838 501 9.839 266 9.840 029 9.840 792 9.841 554 9.842 315 808 807 806 805 804 803 802 801 799 799 798 796 796 794 794 792 792 790 790 788 788 786 786 785 783 783 781 781 780 778 778 777 776 775 774 773 772 771 770 769 769 767 766 766 765 763 763 762 761 760 9.730 014 9.730 589 9.731 162 9.731734 9.732 305 9.732 874 9.733 442 9.734 009 9.734 574 9.735 138 9.735 700 9.736 262 9.736 821 9.737 380 9.737 937 9.738 493 9.739 047 9.739 601 9.740 153 9.740 703 9.741 252 9.741 801 9.742 347 9.742 893 9.743 437 9.743 980 9.744 521 9.745 062 9.745 601 9.746 138 9.746 675 9.747 210 9.747 744 9.748 277 9.748 808 9.749 339 9.749 868 9.750 395 9.750 922 9.751447 9.751971 9.752 494 9.753016 9.753 536 9.754 055 9.754 573 9.755 090 9.755 606 9.756 120 9.756 633 575 573 572 571 569 568 567 565 564 562 562 559 559 557 556 554 554 552 550 549 549 546 546 544 543 541 541 539 537 537 535 534 533 531 531 529 527 527 525 524 523 522 520 519 518 517 516 514 513 512 298 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) 1(>K COMll I log sinh z log tanh z 0.650 0.085 930 0.651 0.086 178 0.652 0.086 427 0.653 0.086 676 0.654 0.086 925 0.655 0.087 175 0.656 0.087 425 0.657 0.087 675 0.658 0.087 926 0.659 0.088 176 0.660 0.088 427 0.661 0.088 679 0.662 0.088 930 0.663 0.089 182 0.664 0.089 434 0.665 0.089 687 0.666 0.089 940 0.667 0.090 193 0.668 0.090 446 0.669 0.090 700 0.670 0.090 954 0.671 0.091 208 0.672 0.091462 0.673 0.091717 0.674 0.091 972 0.675 0.092 227 0.676 0.092 483 0.677 0.092 739 0.678 0.092 995 0.679 0.093 252 0.680 0.093 508 0.681 0.093 765 0.682 0.094 023 0.683 0.094 280 0.684 0.094 538 0.685 0.094 796 0.686 0.095 055 0.687 0.095 313 0.688 0.095 572 0.689 0.095 832 0.690 . 0.096 091 0.691 0.096 351 0.692 0.096 611 0.693 0.096 872 0.694 0.097 132 0.097 393 0.695 0.696 0.097 655 0.697 0.097 916 0.698 0.098 178 0.699 0.098 440 248 249 249 249 250 250 250 251 250 251 252 251 252 252 253 253 253 253 254 254 254 254 255 255 255 256 256 256 257 256 257 258 257 258 258 259 258 259 260 259 260 260 261 260 261 262 261 262 262 262 9.843 075 9.843 835 9.844 593 9.845 351 9.846 107 9.846 863 9.847 618 9.848 372 9.849 125 9.849 877 9.850 628 9.851 379 9.852 128 9.852 877 9.853 625 9.854 372 9.855 118 9.855 863 9.856 608 9.857 352 9.858 094 9.858 836 9.859 578 9.860 318 9.861 057 9.861 796 9.862 534 9.863 271 9.864 007 9.864 743 9.865 477 9.866 211 9.866 944 9.867 676 9.868 408 9.869 138 9.869 868 9.870 597 9.871 325 9.872 053 9.872 779 9.873 505 9.874 230 9.874 955 9.875 678 9.876 401 9.877 123 9.877 844 9.878 565 9.879 285 760 758 758 756 756 755 754 753 752 751 751 749 749 748 747 746 745 745 744 742 742 742 740 739 739 738 737 736 736 734 734 733 732 732 730 730 729 728 728 726 726 725 725 723 723 722 721 721 720 719 9.757 145 9.757 656 9.758 166 9.758 674 9.759 182 9.759 688 9.760 193 9.760 696 9.761 199 9.761 701 9.762 201 9.762 700 9.763 198 9.763 695 9.764 191 9.764 685 9.765 179 9.765 671 9.766 162 9.766 652 9.767 141 9.767 629 9.768 115 9.768 601 9.769 085 9.769 569 9.770 051 9.770 532 9.771012 9.771491 9.771 969 9.772 446 9.772 921 9.773 396 9.773 870 9.774 342 9.774 813 9.775 284 9.775 753 9.776 221 9.776 688 9.777 154 9.777 619 9.778 083 9.778 546 9.779 008 9.779 469 9.779 928 9.780 387 9.780 845 511 510 508 508 506 505 503 503 502 500 499 498 497 496 494 494 492 491 490 489 488 486 486 484 484 482 481 480 479 478 477 475 475 474 472 471 471 469 468 467 466 465 464 463 462 461 459 459 458 456 APPENDIX 299 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh x log tanh x 0.700 0.701 0.702 0.703 0.704 0.705 0.706 0.707 0.708 0.709 0.710 0.711 0.712 0.713 0.714 0.715 0.716 0.717 0.718 0.719 0.720 0.721 0.722 0.723 0.724 0.725 0.726 0.727 0.728 0.729 0.730 0.731 0.732 0.733 0.734 0.735 0.736 0.737 0.738 0.739 0.740 0.741 0.742 0.743 0.744 0.745 0.746 0.747 0.748 0.749 0.098 702 0.098 965 0.099 228 0.099 491 0.099 754 0.100 018 0.100 282 0.100 546 0.100 811 0.101076 0.101341 0.101606 0.101872 0.102 138 0.102 404 0.102 670 0.102 937 0.103 204 0.103 471 0.103 739 0.104 006 0.104 274 0.104 543 0.104 811 0.105 080 0.105 349 0.105 619 0.105 888 0.106 158 0.106 429 0.106 699 0.106 970 0.107 241 0.107 512 0.107 783 0.108 055 0.108 327 0.108 600 0.108 872 0.109 145 0.109 418 0.109 691 0.109 965 0.110 239 0.110 513 0.110 788 0.111062 0.111337 0.111612 0.111888 263 263 263 263 264 264 264 265 265 265 265 266 266 266 266 267 267 267 268 267 268 269 268 269 269 270 269 270 271 270 271 271 271 271 272 272 273 272 273 273 273 274 274 274 275 274 275 275 276 276 9.880 004 9.880 722 9.881 439 9.882 156 9.882 872 9.883 587 9.884 302 9.885 015 9.885 728 9.886 441 9.887 152 9.887 863 9.888 573 9.889 282 9.889 991 9.890 699 9.891 406 9.892 113 9.892 818 9.893 523 9.894 228 9.894 931 9.895 634 9.896 336 9.897 038 9.897 739 9.898 439 9.899 138 9.899 837 9.900 535 9.901233 9.901 929 9.902 625 9.903 321 9.904 015 9.904 709 9.905 402 9.906 095 9.906 787 9.907 478 9.908 169 9.908 859 9.909 548 9.910 237 9.910 925 9.911 612 9.912 299 9.912 985 9.913 670 9.914 355 718 717 717 716 715 715 713 713 713 711 711 710 709 709 708 707 707 705 705 705 703 703 702 702 701 700 699 699 698 698 696 696 696 694 694 693 693 692 691 691 690 689 689 688 687 687 686 685 685 684 9.781 301 9.781 757 9.782 211 9.782 665 9.783 118 9.783 569 9.784 020 9.784 469 9.784 918 9.785 365 9.785 811 9.786 257 9.786 701 9.787 145 9.787 587 9.788 029 9.788 469 9.788 909 9.789 347 9.789 785 9.790 221 9.790 657 9.791 091 9.791 525 9.791958 9.792 389 9.792 820 9.793 250 9.793 679 9.794 107 9.794 534 9.794 960 9.795 385 9.795 809 9.796 232 9.796 654 9.797 075 9.797 496 9.797 915 9.798 333 9.798 751 9.799 167 9.799 583 9.799 998 9.800 412 9.800 825 9.801 237 9.801648 9.802 058 9.802 467 456 454 454 453 451 451 449 449 447 446 446 444 444 442 442 440 440 438 438 436 436 434 434 433 431 431 430 429 428 427 426 425 424 423 422 421 421 419 418 418 416 416 415 414 413 412 411 410 409 409 300 AIMMADIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh i i log sinh z log tanh i i 0.750 0.751 0.752 0.753 0.754 0.755 0.756 0.757 0.758 0.759 0.760 0.761 0.762 0.763 0.764 0.765 0.766 0.767 0.768 0.769 0.770 0.771 0.772 0.773 0.774 0.775 0.776 0.777 0.778 0.779 0.780 0.781 .782 .783 .784 .785 .786 0.787 0.788 0.789 0.790 0.791 0.792 0.793 0.794 0.795 0.796 0.797 0.798 0.799 0.112 164 0.112 440 0.112 716 0.112 992 0.113 269 0.113 546 0.113 823 0.114 101 0.114 379 0.114 657 0.114 935 0.115 213 0.115 492 0.115 771 0.116 051 0.116 330 0.116 610 0.116 890 0.117 170 0.117 451 0.117 732 0.118 013 0.118 294 0.118 576 0.118 858 0.119 140 0.119 422 0.119 705 0.119 987 0.120 271 0.120 554 0.120 837 0.121 121 0.121 405 0.121690 0.121974 0.122 259 0.122 544 0.122 830 0.123 115 0.123 401 0.123 687 0.123 973 0.124 260 0.124 547 0.124 834 0.125 121 0.125 409 0.125 696 0.125 984 276 276 276 277 277 277 278 278 278 278 278 279 279 280 279 280 280 280 281 281 281 281 282 282 282 282 283 282 284 283 283 284 284 285 284 285 285 286 285 286 286 286 287 287 287 287 288 287 288 289 9.915 039 9.915 723 9.916 405 9.917 088 9.917 769 9.918 450 9.919 130 9.919 810 9.920 489 9.921 167 9.921 845 9.922 522 9.923 199 9.923 875 9.924 550 9.925 225 9.925 899 9.926 572 9.927 245 9.927 917 9.928 589 9.929 260 9.929 930 9.930 600 9.931269 9.931938 9.932 606 9.933 273 9.933 940 9.934 606 9.935 272 9.935 937 9.936 602 9.937 265 9.937 929 9.938 592 9.939 254 9.939 915 9.940 576 9.941 237 9.941897 9.942 556 9.943 215 9.943 873 9.944 531 9.945 188 9.945 844 9.946 500 9.947 156 9.947 810 684 682 683 681 681 680 680 679 678 678 677 677 676 675 675 674 673 673 672 672 671 670 670 669 669 668 667 667 666 666 665 665 663 664 663 662 661 661 661 660 659 659 658 658 657 656 656 656 654 655 9.802 876 9.803 283 9.803 690 9.804 095 9.804 500 9.804 904 9.805 307 9.805 709 9.806 110 9.806 511 9.806 910 9.807 309 9.807 706 9.808 103 9.808 499 9.808 894 9.809 289 9.809 682 9.810 074 9.810 466 9.810 857 9.811 247 9.811636 9.812 024 9.812 412 9.812 798 9.813 184 9.813 569 9.813 953 9.814 336 9.814 718 9.815 100 9.815 480 9.815 860 9.816 239 9.816 617 9.816 995 9.817 371 9.817 747 9.818 122 9.818 496 9.818 869 9.819 241 9.819 613 9.819 984 9.820 354 9.820 723 9.821 092 9.821 459 9.821 826 407 407 405 405 404 403 402 401 401 399 399 397 397 396 395 395 393 392 392 391 390 389 388 388 386 386 385 384 383 382 382 380 380 379 378 378 376 376 375 374 373 372 372 371 370 369 369 367 367 366 APPENDIX 301 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh x log tanh x 0.800 0.801 0.802 0.803 0.804 0.805 0.806 0.807 0.808 0.809 0.810 0.811 0.812 0.813 0.814 0.815 0.816 0.817 0.818 0.819 0.820 0.821 0.822 0.823 0.824 0.825 0.826 0.827 0.828 0.829 0.830 0.831 0.832 0.833 0.834 0.835 0.836 0.837 0.838 0.839 0.840 0.841 0.842 0.843 0.844 0.845 0.846 0.847 0.848 0.849 0.126 273 0.126 561 0.126 850 0.127 139 0.127 428 0.127 718 0.128 007 0.128 297 0.128 588 0.128 878 0.129 169 0.129 460 0.129 751 0.130 042 0.130 334 0.130 626 0.130 918 0.131210 0.131503 0.131795 0.132 089 0.132 382 0.132 675 0.132 969 0.133 263 0.133 557 0.133 852 0.134 147 0.134 442 0.134 737 0.135 032 0.135 328 0.135 624 0.135 920 0.136 216 0.136 513 0.136 809 0.137 106 0.137 404 0.137 701 0.137 999 0.138 297 0.138 595 0.138 894 0.139 192 0.139 491 0.139 790 0.140 090 0.140 389 0.140 689 288 289 289 289 290 289 290 291 290 291 291 291 291 292 292 292 292 293 292 294 293 293 294 294 294 295 295 295 295 295 296 296 296 296 297 296 297 298 297 298 298 298 299 298 299 299 300 299 300 300 9.948 465 9.949 119 9.949 772 9.950 424 9.951076 9.951 728 9.952 379 9.953 030 9.953 679 9.954 329 9.954 978 9.955 626 9.956 274 9.956 921 9.957 568 9.958 214 9.958 860 9.959 505 9.960 150 9.960 794 9.961 437 9.962 080 9.962 723 9.963 365 9.964 006 9.964 647 9.965 288 9.965 928 9.966 567 9.967 206 9.967 845 9.968 483 9.969 120 9.969 757 9.970 394 9.971030 9.971 665 9.972 300 9.972 935 9.973 569 9.974 202 9.974 835 9.975 468 9.976 100 9.976-731 9.977 362 9.977 993 9.978 623 9.979 253 9.979 882 654 653 652 652 652 651 651 649 650 649 648 648 647 647 646 646 645 645 644 643 643 643 642 641 641 641 640 639 639 639 638 637 637 637 636 635 635 635 634 633 633 633 632 631 631 631 630 630 629 629 9.822 192 9.822 557 9.822 922 9.823 285 9.823 648 9.824 010 9.824 372 9.824 732 9.825 092 9.825 451 9.825 809 9.826 167 9.826 523 9.826 879 9.827 234 9.827 588 9.827 942 9.828 295 9.828 647 9.828 998 9.829 349 9.829 699 9.830 048 9.830 396 9.830 743 9.831090 9.831 436 9.831 781 9.832 126 9.832 470 9.832 813 9.833 155 9.833 497 9.833 838 9.834 178 9.834 517 9.834 856 9.835 194 9.835 531 9.835 867 9.836 203 9.836 538 9.836 873 9.837 206 9.837 539 9.837 871 9.838 203 9.838 534 9.838 864 9.839 193 365 365 363 363 362 362 360 360 359 358 358 356 356 355 354 354 353 352 351 351 350 349 348 347 347 346 345 345 344 343 342 342 341 340 339 339 338 337 336 336 335 335 333 333 332 332 331 330 329 329 302 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) 0.850 0.851 0.852 0.853 0.854 0.855 0.856 0.857 0.858 0.859 0.860 0.861 0.862 0.863 0.864 0.865 0.S66 0.867 0.868 0.869 0.870 0.871 0.872 0.873 0.874 0.875 0.876 0.877 0.878 0.879 0.880 0.881 0.882 0.883 0.884 0.885 0.886 0.887 0.888 0.889 0.890 0.891 0.892 0.893 0.894 0.895 0.896 0.897 0.898 0.899 log cosh i 0.140 989 0.141289 0.141590 0.141890 0.142 191 0.142 492 0.142 794 0.143 095 0.143 397 0.143 699 0.144 001 0.144 304 0.144 607 0.144 910 0.145 213 0.145 516 0.145 820 0.146 124 0.146 428 0.146 732 0.147 036 0.147 341 0.147 646 0.147 951 0.148 257 0.148 562 0.148 868 0.149 174 0.149 480 0.149 787 0.150 093 0.150 400 0.150 707 0.151015 0.151322 0.151 630 0.151 938 0.152 246 0.152 555 0.152 863 0.153 172 0.153 481 0.153 790 0.154 100 0.154 410 0.154 720 0.155 030 0.155 340 0.155 651 0.155 961 300 301 300 301 301 302 301 302 302 302 303 303 303 303 303 304 304 304 304 304 305 305 305 306 305 306 306 306 307 306 307 307 308 307 308 308 308 309 308 309 309 309 310 310 310 310 310 311 310 311 log sinh x 9.980 511 9.981 139 9.981 766 9.982 394 9.983 021 9.983 647 9.984 273 9.984 898 9.985 523 9.986 147 9.986 771 9.987 395 9.988 018 9.988 641 9.989 263 9.989 884 9.990 506 9.991 126 9.991 747 9.992 367 9.992 986 9.993 605 9.994 224 9.994 842 9.995 459 9.996 077 9.996 693 9.997 310 9.997 926 9.998 541 9.999 156 9.999 771 0.000 385 0.000 998 0.001 612 0.002 224 0.002 837 0.003 449 0.004 060 0.004 671 0.005 282 0.005 892 0.006 502 0.007 112 Q.007 721 0.008 329 0.008 938 0.009 545 0.010 153 0.010 760 628 627 628 627 626 626 625 625 624 624 624 623 623 622 621 622 620 621 620 619 619 619 618 617 618 616 617 616 615 615 615 614 613 614 612 613 612 611 611 611 610 610 610 609 608 609 607 608 607 606 log t.-illll z 9.839 522 9.839 850 9.840 177 9.840 503 9.840 829 9.841 155 9.841 479 9.841 803 9.842 126 9.842 448 9.842 770 9.843 091 9.843 411 9.843 731 9.844 050 9.844 368 9.844 686 9.845 003 9.845 319 9.845 635 9.845 950 9.846 264 9.846 578 9.846 891 9.847 203 9.847 514 9.847 825 9.848 136 9.848 445 9.848 754 9.849 063 9.849 370 9.849 677 9.849 984 9.850 289 9.850 594 9.850 899 9.851 203 9.851506 9.851 808 9.852 110 9.852 411 9.852 712 9.853 012 9.853 311 9.853 610 9.853 908 9.854 205 9.854 502 9.854 798 328 327 326 326 326 324 324 323 322 322 321 320 320 319 318 318 317 316 316 315 314 314 313 312 311 311 311 309 309 309 307 307 307 305 305 305 304 303 302 302 301 301 300 299 299 298 297 297 296 296 APPENDIX 303 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh x log tanh x 0.900 0.901 0.902 0.903 0.904 0.905 0.906 0.907 0.908 0.909 0.910 0.911 0.912 0.913 0.914 0.915 0.916 0.917 0.918 0.919 0.920 0.921 0.922 0.923 0.924 0.925 0.926 0.927 0.928 0.929 0.930 0.931 0.932 0.933 0.934 0.935 0.936 0.937 0.938 0.939 0.940 0.941 0.942 0.943 0.944 0.945 0.946 0.947 0.948 0.949 0.156 272 0.156 584 0.156 895 0.157 207 0.157 518 0.157 830 0.158 143 0.158 455 0.158 768 0.159 081 0.159 394 0.159 707 0.160 021 0.160 334 0.160 648 0.160 962 0.161277 0.161591 0.161906 0.162 221 0.162 536 0.162 851 0.163 167 0.163 483 0.163 799 0.164 115 0.164 431 0.164 748 0.165 065 0.165 382 0.165 699 0.166 016 0.166 334 0.166 651 0.166 969 0.167 288 0.167 606 0.167 925 0.168 243 0.168 563 0.168 882 0.169 201 0.169 521 0.169 841 0.170 161 0.170 481 0.170 801 0.171 122 0.171442 0.171763 312 311 312 311 312 313 312 313 313 313 313 314 313 314 314 315 314 315 315 315 315 316 316 316 316 316 317 317' 317 317 317 318 317 318 319 318 319 318 320 319 319 320 320 320 320 320 321 320 321 322 0.011 366 0.011 972 0.012 578 0.013 183 0.013 788 0.014 392 0.014 996 0.015 600 0.016 203 0.016 806 0.017 409 0.018 011 0.018 612 0.019 214 0.019 814 0.020 415 0.021 015 0.021 615 0.022 214 0.022 813 0.023 411 0.024 009 0.024 607 0.025 204 0.025 801 0.026 398 0.026 994 0.027 590 0.028 185 0.028 780 0.029 375 0.029 969 0.030 563 0.031 156 0.031 749 0.032 342 0.032 935 0.033 527 0.034 118 0.034 709 0.035 300 0.035 891 0.036 481 0.037 071 0.037 660 0.038 249 0.038 838 0.039 426 0.040 014 0.040 602 606 606 605 605 604 604 604 603 603 603 602 601 602 600 601 600 600 599 599 598 598 598 597 597 597 596 596 595 595 595 594 594 593 593 593 593 592 591 591 591 591 590 590 589 589 589 588 588 588 587 9.855 094 9.855 389 9.855 683 9.855 977 9.856 270 9.856 562 9.856 854 9.857 145 9.857 436 9.857 726 9.858 015 9.858 304 9.858 592 9.858 879 9.859 166 9.859 453 9.859 738 9.860 023 9.860 308 9.860 592 9.860 875 9.861 158 9.861 440 9.861 722 9.862 003 9.862 283 9.862 563 9.862 842 9.863 121 9.863 399 9.863 676 9.863 953 9.864 229 9.864 505 9.864 780 9.865 055 9.865 328 9.865 602 9.865 875 9.866 147 9.866 419 9.866 690 9.866 960 9.867 230 9.867 500 9.867 769 9.868 037 9.868 305 9.868 572 9.868 838 295 294 294 293 292 292 291 291 290 289 289 288 287 287 287 285 285 285 284 283 283 282 282 281 280 280 279 279 278 277 277 276 276 275 275 273 274 273 272 272 271 270 270 270 269 268 268 267 266 266 304 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) leg nak i log sinh z log tanh z 0.960 0.951 0.952 0.953 0.954 0.955 0.956 0.957 0.958 0.959 0.960 0.961 0.962 0.963 0.964 0.965 0.966 0.967 0.968 0.969 0.970 0.971 0.972 0.973 0.974 0.975 0.976 0.977 0.978 0.979 0.980 0.981 0.982 0.983 0.984 0.985 0.988 0.989 0.990 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 987 0.172 085 0.172 406 0.172 728 0.173 049 0.173 371 0.173 694 0.174 016 0.174 338 0.174 661 0.174 984 0.175 307 0.175 631 0.175 954 0.176 278 0.176 602 0.176 926 0.177 250 0.177 575 0.177 899 0.178 224 0.178 549 0.178 875 0.179 200 0.179 526 0.179 851 0.180 177 0.180 504 0.180 830 0.181 157 0.181483 0.181810 0.182 137 0.182 465 0.182 792 0.183 120 0.183 448 0.183 776 0.184 104 0.184 433 0.184 761 0.185 090 0.185 419 0.185 748 0.186 078 0.186 407 0.186 737 0.187 067 0.187 397 0.187 727 0.188 058 321 322 321 322 323 322 322 323 323 323 324 323 324 324 324 324 325 324 325 325 326 325 326 325 326 327 326 327 326 327 327 328 327 328 328 328 328 329 328 329 329 329 330 329 330 330 330 330 331 331 0.041 189 0.041 776 0.042 362 0.042 949 0.043 534 0.044 120 0.044 705 0.045 290 0.045 874 0.046 458 0.047 042 0.047 625 0.048 208 0.048 791 0.049 373 0.049 955 0.050 537 0.051 118 0.051 699 0.052 279 0.052 860 0.053 439 0.054 019 0.054 598 0.055 177 0.055 756 0.056 334 0.056 912 0.057 489 0.058 066 0.058 643 0.059 220 0.059 796 0.060 372 0.060 947 0.061 523 0.062 098 0.062 672 0.063 246 0.063 820 0.064 394 0.064 967 0.065 540 0.066 113 0.066 685 0.067 257 0.067 829 0.068 400 0.068 971 0.069 542 587 5N6 587 585 586 585 585 584 584 584 583 583 583 582 582 582 581 581 580 581 579 580 579 579 579 578 578 577 577 577 577 576 576 575 576 575 574 574 574 574 573 573 573 572 572 572 571 571 571 570 9.869 104 9.869 370 9.869 635 9.869 899 9.870 163 9.870 426 9.870 689 9.870 951 9.871 213 9.871 474 9.871 735 9.871995 9.872 254 9.872 513 9.872 771 9.873 029 9.873 286 9.873 543 9.873 799 9.874 055 9.874 310 9.874 565 9.874 819 9.875 073 9.875 326 9.875 578 9.875 830 9.876 082 9.876 333 9.876 583 9.876 833 9.877 082 9.877 331 9.877 580 9.877 827 9.878 075 9.878 322 9.878 568 9.878 814 9.879 059 9.879 304 9.879 548 9.879 792 9.880 035 9.880 278 9.880 520 9.880 762 9.881 003 9.881 244 9.881484 266 265 264 264 263 263 262 262 261 261 260 259 259 258 258 257 257 256 256 255 255 254 254 253 252 252 252 251 250 250 249 249 249 247 248 247 246 246 245 245 244 244 243 243 242 242 241 241 240 240 APPENDIX 305 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh i log sinh x log tanh x 1.000 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1.009 1.010 1.011 1.012 1.013 1.014 1.015 1.016 1.017 1.018 1.019 1.020 1.021 1.022 1.023 1.024 1.025 1.026 1.027 1.028 1.029 1.030 1.031 1.032 1.033 1.034 1.035 1.036 1.037 1.038 1.039 1.040 1.041 1.042 1.043 1.044 1.045 1.046 1.047 1.048 1.049 0.188 389 0.188 719 0.189 050 0.189 382 0.189 713 0.190 045 0.190 376 0.190 708 0.191040 0.191373 0.191705 0.192 038 0.192 371 0.192 704 0.193 037 0.193 370 0.193 704 0.194 038 0.194 372 0.194 706 0.195 040 0.195 374 0.195 709 0.196 044 0.196 379 0.196 714 0.197 049 0.197 385 0.197 720 0.198 056 0.198 392 0.198 728 0.199 065 0.199 401 0.199 738 0.200 075 0.200 412 0.200 749 0.201 087 0.201 424 0.201 762 0.202 100 0.202 438 0.202 776 0.203 115 0.203 453 0.203 792 0.204 131 0.204 470 0.204 809 330 331 332 331 332 331 332 332 333 332 333 333 333 333 333 334 334 334 334 334 334 335 335 335 335 335 336 335 336 336 336 337 336 337 337 337 337 338 337 338 338 338 338 339 "338 339 339 339 339 340 0.070 112 0.070 682 0.071 252 0.071 822 0.072 391 0.072 960 0.073 528 0.074 096 0.074 664 0.075 232 0.075 799 0.076 366 0.076 933 0.077 499 0.078 065 0.078 631 0.079 196 0.079 762 0.080 326 0.080 891 0.081455 0.082 019 0.082 583 0.083 146 0.083 709 0.084 272 0.084 835 0.085 397 0.085 959 0.086 520 0.087 082 0.087 643 0.088 203 0.088 764 0.089 324 0.089 884 0.090 443 0.091 003 0.091 562 0.092 121 0.092 679 0.093 237 0.093 795 0.094 353 0.094 910 0.095 467 0.096 024 0.096 580 0.097 136 0.097 692 570 570 570 569 569 568 568 568 568 567 567 567 566 566 566 565 566 564 565 564 564 564 563 563 563 563 562 562 561 562 561 560 561 560 560 559 560 559 559 558 558 558 558 557 557 557 556 556 556 556 9.881724 9.881963 -9.882 202 9.882 440 9.882 678 9.882 915 9.883 152 9.883 388 9.883 624 9.883 859 9.884 094 9.884 328 9.884 562 9.884 795 9.885 028 9.885 261 9.885 493 9.885 724 9.885 955 9.886 185 9.886 415 9.886 645 9.886 874 9.887 103 9.887 331 9.887 558 9.887 786 9.888 012 9.888 238 9.888 464 9.888 690 9.888 914 9.889 139 9.889 363 9.889 586 9.889 809 9.890 032 9.890 254 9.890 475 9.890 696 9.890 917 9.891 137 9.891357 9.891 576 9.891795 9.892 014 9.892 232 9.892 449 9.892 666 9.892 883 239 239 238 238 237 237 236 236 235 235 234 234 233 233 233 232 231 .231 230 230 230 229 229 228 227 228 226 226 226 226 224 225 224 223 223 223 222 221 221 221 220 220 219 219 219 218 217 217 217 216 306 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh x log tanh z 1.050 1.051 1.052 1.053 1.054 1.055 1.056 1.057 1.058 1.059 1.060 1.061 1.062 1.063 1.064 1.065 1.066 1.067 1.068 1.069 1.070 1.071 1.072 1.073 1.074 1.075 1.076 1.077 1.078 1.079 1.080 1.081 1.082 1.083 1.084 1.085 1.086 1.087 1.088 1.089 1.090 1.091 1.092 1.093 1.094 1.095 1.096 1.097 1.098 1.099 0.205 149 0.205 488 0.205 828 0.206 168 0.206 508 0.206 848 0.'207 189 0.207 530 0.207 870 0.208 211 0.208 552 0.208 894 0.209 235 0.209 577 0.209 919 0.210 261 0.210 603 0.210 945 0.211 287 0.211 630 0.211 973 0.212 316 0.212 659 0.213 002 0.213 346 0.213 689 0.214 033 0.214 377 0.214 721 0.215 065 0.215 410 0.215 754 0.216 099 0.216 444 0.216 789 0.217 134 0.217 479 0.217 825 0.218 171 0.218 516 0.218 862 0.219 209 0.219 555 0.219 901 0.220 248 0.220 595 0.220 942 0.221 289 0.221 636 0.221 983 339 340 340 340 340 341 341 340 341 341 342 341 342 342 342 342 342 342 343 343 343 343 343 344 343 344 344 344 344 345 344 345 345 345 345 345 346 346 345 346 347 346 346 347 347 347 347 347 347 348 0.098 248 0.098 803 0.099 358 0.099 913 0.100 468 0.101 022 0.101576 0.102 130 0.102 683 0.103 236 0.103 789 0.104 342 0.104 894 0.105 446 0.105 998 0.106 550 0.107 101 0.107 652 0.108 203 0.108 753 0.109 304 0.109 854 0.110 403 0.110 953 0.111502 0.112 051 0.112 600 0.113 148 0.113 696 0.114 244 0.114 792 0.115 339 0.115 886 0.116 433 0.116 980 0.117 526 0.118 072 0.118 618 0.119 164 0.119 709 0.120 254 0.120 799 0.121344 0.121888 0.122 432 0.122 976 0.123 520 0.124 063 0.124 606 0.125 149 555 555 555 555 554 554 554 553 553 553 553 552 552 552 552 551 551 551 550 551 550 549 550 549 549 549 548 548 548 548 547 547 547 547 546 546 546 546 545 545 545 545 544 544 544 544 543 543 543 543 9.893 099 9.893 315 9.893 530 9.893 745 9.893 960 9.894 173 9.894 387 9.894 600 9.894 813 9.895 025 9.895 237 9.895 448 9.895 659 9.895 870 9.896 080 9.896 289 9.896 498 9.896 707 9.896 915 9.897 123 9.897 331 9.897 538 9.897 744 9.897 951 9.898 156 9.898 362 9.898 567 9.898 771 9.898 975 9.899 179 9.899 382 9.899 585 9.899 787 9.899 989 9.900 191 9.900 392 9.900 593 9.900 793 9.900 993 9.901 193 9.901 392 9.901 591 9.901 789 9.901 987 9.902 184 9.902 381 9.902 578 9.902 774 9.902 970 9.903 166 216 215 215 215 213 214 213 213 212 212 211 211 211 210 209 209 209 208 208 208 207 206 207 205 206 205 204 204 204 203 203 202 202 202 201 201 200 200 200 199 199 198 198 197 197 197 196 196 196 195 APPENDIX 307 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) 1.100 1.101 1.102 1.103 1.104 1.105 1.106 1.107 1.108 1.109 1.110 1.111 1.112 113 114 115 116 1.117 1.118 1.119 1.120 1.121 1.122 1.123 1.124 1.125 1.126 1.127 1.128 1.129 1.130 1.131 1.132 1.133 1.134 1.135 1.136 1.137 1.138 1.139 1.140 1.141 1.142 1.143 1.144 145 146 147 148 149 log cosh i 0.222 331 0.222 679 0.223 027 0.223 375 0.223 723 0.224 071 0.224 420 0.224 768 0.225 117 0.225 466 0.225 815 0.226 165 0.226 514 0.226 864 0.227 213 0.227 563 0.227 913 0.228 263 0.228 614 0.228 964 0.229 315 0.229 666 0.230 017 0.230 368 0.230 719 0.231070 0.231422 0.231 774 0.232 126 0.232 478 0.232 830 0.233 182 0.233 534 0.233 887 0.234 240 0.234 593 0.234 946 0.235 299 0.235 652 0.236 006 0.236 359 0.236 713 0.237 067 0.237 421 0.237 775 0.238130 0.238 484 0.238 839 0.239 194 0.239 548 348 348 348 348 348 349 348 349 349 349 350 349 350 349 350 350 350 351 350 351 351 351 351 351 351 352 352 352 352 352 352 352 353 353 353 353 353 353 354 353 354 354 354 354 355 354 355 355 354 356 log sinh x 0.125 692 0.126 234 0.126 776 0.127 318 0.127 860 0.128 401 0.128 943 0.129 484 0.130 024 0.130 565 0.131 105 0.131645 0.132 185 0.132 724 0.133 264 0.133 803 0.134 342 0.134 880 0.135 419 0.135 957 0.136 495 0.137 032 0.137 570 0.138 107 0.138 644 0.139 181 0.139 717 0.140 253 0.140 789 0.141325 0.141861 0.142 396 0.142 931 0.143 466 0.144 001 0.144 536 0.145 070 0.145 604 0.146 138 0.146 671 0.147 205 0.147 738 0.148 271 0.148 803 0.149 336 0.149 868 0.150 400 0.150 932 0.151464 0.151995 542 542 542 542 541 542 541 540 541 540 540 540 539 540 539 539 538 539 538 538 537 538 537 537 537 536 536 536 536 536 535 535 535 535 535 534 534 534 533 534 533 533 532 533 532 532 532 532 531 531 log tanh x 9.903 361 9.903 555 9.903 750 9.903 944 9.904 137 9.904 330 9.904 523 9.904 715 9.904 907 9.905 099 9.905 290 9.905 480 9.905 671 9.905 861 9.906 050 9.906 240 9.905 428 9.906 617 9.906 805 9.906 992 9.907 180 9.907 366 9.907 553 9.907 739 9.907 925 9.908 110 9.908 295 9.908 480 9.908 664 9.908 848 9.909 031 9.909 214 9.909 397 9.909 579 9.909 761 9.909 943 9.910 124 9.910 305 9.910 485 9.910 666 9.910 845 9.911025 9.911 204 9.911 382 9.911561 9.911 739 9.911916 9.912 093 9.912 270 9.912 447 194 195 194 193 193 193 192 192 192 191 190 191 190 189 190 188 189 188 187 188 186 187 186 186 185 185 185 184 184 183 183 183 182 182 182 181 181 180 181 179 180 179 178 179 178 177 177 177 177 176 :*os APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) 1.150 1.151 1.152 1.153 1.154 1.155 1.156 1.157 1.158 1.159 1.160 1.161 1.162 1.163 1.164 1.165 1.166 1.167 1.168 1.169 1.170 1.171 1.172 1.173 1.174 1.175 1.176 1.177 1.178 1.179 1.180 1.181 1.182 1.183 1.184 1.185 1.186 1.187 1.188 1.189 .190 .191 .192 .193 .194 .195 1.196 1.197 1.198 1.199 log cosh z 0.239 904 0.240 259 0.240 614 0.240 970 0.241325 0.241 681 0.242 037 0.242 393 0.242 749 0.243 106 0.243 462 0.243 819 0.244 176 0.244 532 0.244 890 0.245 247 0.245 604 0.245 962 0.246 319 0.246 677 0.247 035 0.247 393 0.247 751 0.248109 0.248 468 0.248 827 0.249 185 0.249 544 0.249 903 0.250 262 0.250 622 0.250 981 0.251341 0.251 700 0.252 060 0.252 420 0.252 780 0.253 140 0.253 501 0.253 861 0.254 222 0.254 583 0.254 944 0.255 305 0.255 666 0.256 027 0.256 389 0.256 750 0.257112 0.257 474 355 355 356 355 356 356 356 356 357 356 357 357 356 358 357 357 358 357 358 358 358 358 358 359 359 358 359 359 359 360 359 360 359 360 360 360 360 361 360 361 361 361 361 361 361 362 361 362 362 362 log sinh z 0.152 526 0.153 057 0.153 588 0.154 119 0.154 649 0.155 179 0.155 709 0.156 239 0.156 768 0.157 297 0.157 826 0.158 355 0.158 884 0.159 412 0.159 941 0.160 469 0.160 996 0.161524 0.162 051 0.162 579 0.163 106 0.163 632 0.164 159 0.164 685 0.165 211 0.165 737 0.166 263 0.166 789 0.167 314 0.167 839 0.168 364 0.168 889 0.169 414 0.169 938 0.170 462 0.170 986 0.171510 0.172 033 0.172 557 0.173 080 0.173 603 0.174 126 0.174 648 0.175 171 0.175 693 0.176 215 0.176 737 0.177 258 0.177 780 0.178 301 531 531 531 530 530 530 530 529 529 529 529 529 528 529 528 527 528 527 528 527 526 527 526 526 526 526 526 525 525 525 525 525 524 524 524 524 523 524 523 523 523 522 523 522 522 522 521 522 521 521 log tanh x 9.912 623 9.912 798 9.912 974 9.913 149 9.913 324 9.913 498 9.913 672 9.913 846 9.914 019 9.914 192 9.914 364 9.914 536 9.914 708 9.914 880 9.915 051 9.915 222 9.915 392 9.915 562 9.915 732 9.915 902 9.916 071 9.916 239 9.916 408 9.916 576 9.916 744 9.916 911 9.917 078 9.917 245 9.917 411 9.917 577 9.917 743 9.917 908 9.918 073 9.918 238 9.918 402 9.918 566 9.918 730 9.918 893 9.919 056 9.919 219 9.919 381 9.919 543 9.919 705 9.919 866 9.920 027 9.920 188 9.920 348 9.920 508 9.920 668 9.920 827 7li 75 75 74 71 71 73 73 72 72 72 72 71 71 70 70 70 70 00 08 00 os OS 07 07 07 00 00 00 05 05 05 04 04 01 03 03 03 02 02 02 01 (11 01 00 00 00 50 59 APPENDIX 309 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh x log tanh z 1.200 1.201 1.202 1.203 1.204 1.205 1.206 1.207 1.208 1.209 .210 .211 .212 .213 .214 .215 1.216 1.217 1.218 1.219 1.220 1.221 1.222 1.223 1.224 1.225 1.226 1.227 1.228 1.229 1.230 1.231 1.232 1.233 1.234 1.235 1.236 1.237 1.238 1.239 1.240 1.241 1.242 1.243 1.244 1.245 1.246 1.247 1.248 1.249 0.257 836 0.258 198 0.258 560 0.258 923 0.259 285 0.259 648 0.260 011 0.260 373 0.260 736 0.261 100 0.261463 0.261 826 0.262 190 0.262 554 0.262 917 0.263 281 0.263 645 0.264 010 0.264 374 0.264 738 0.265 103 0.265 468 0.265 833 0.266 198 0.266 563 0.266 928 0.267 293 0.267 659 0.268 024 0.268 390 0.268 756 0.269 122 0.269 488 0.269 854 0.270 221 0.270 587 0.270 954 0.271 321 0.271 687 0.272 054 0.272 422 0.272 789 0.273 156 0.273 524 0.273 891 0.274 259 0.274 627 0.274 995 0.275 363 0.275 731 362 362 363 362 363 363 362 363 364 363 363 364 364 363 364 364 365 364 364 365 365 365 365 365 365 365 366 365 366 366 366 366 366 367 366 367 367 366 367 368 367 367 368 367 368 368 368 368 368 368 0.178 822 0.179 343 0.179 864 0.180 384 0.180 904 0.181424 0.181944 0.182 464 0.182 984 0.183 503 0.184 022 0.184 541 0.185 060 0.1S5 578 0.186 097 0.186 615 0.187 133 0.187 651 0.188 169 0.188 686 0.189 204 0.189 721 0.190 238 0.190 754 0.191271 0.191787 0.192 304 0.192 820 0.193 336 0.193 851 0.194 367 0.194 882 0.195 397 0.195 912 0.196 427 0.196 942 0.197 456 0.197 971 0.198 485 0.198 999 0.199 512 0.200 026 0.200 539 0.201053 0.201 566 0.202 079 0.202 591 0.203 104 0.203 616 0.204129 521 521 520 520 520 520 520 520 519 519 519 519 518 519 518 518 518 518 517 518 517 517 516 517 516 517 516 516 515 516 515 515 515 515 515 514 515 514 514 513 514 513 514 513 513 512 513 512 513 512 9.920 986 9.921 145 9.921 303 9.921 461 9.921 619 9.921 777 9.921 934 9.922 091 9.922 247 9.922 403 9.922 559 9.922 715 9.922 870 9.923 025 9.923 179 9.923 334 9.923 488 9.923 641 9.923 795 9.923 948 9.924 100 9.924 253 9.924 405 9.924 557 9.924 708 9.924 859 9.925 010 9.925 161 9.925 311 9.925 461 9.925 611 9.925 760 9.925 909 9.926 058 9.926 206 9.926 355 9.926 502 9.926 650 9.926 797 9.926 944 9.927 091 9.927 237 9.927 383 9.927 529 9.927 674 9.927 820 9.927 965 9.928 109 9.928 253 9.928 397 59 .58 58 58 58 57 57 56 56 56 56 55 55 54 55 54 53 54 53 52 53 52 52 51 51 51 51 50 50 50 49 49 49 48 49 47 48 47 47 47 46 46 46 45 48 45 44 44 44 44 310 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS {Continued) !<>>• cosh i log sinh z log tanh z 1.250 1.251 1.252 1.253 1.254 1.255 1^56 1.257 1.258 1.259 1.260 1.261 1.262 1.263 1.264 1.265 1.266 1.267 1.268 1.269 1.270 1.271 1.272 1.273 1.274 1.275 1.276 1.277 1.278 1.279 1.280 1.281 1.282 1.283 1.284 1.285 1.286 1.287 1.288 1.289 1.290 1.291 1.292 1.293 1.294 1.295 1.296 1.297 1.298 1.299 0.276 099 0.276 468 0.276 837 0.277 205 0.277 574 0.277 943 0.278 312 0.278 681 0.279 051 0.279 420 0.279 790 0.280 159 0.280 529 0.280 899 0.281 269 0.281 639 0.282 009 0.282 380 0.282 750 0.283 121 0.283 492 0.283 863 0.284 233 0.284 605 0.284 976 0.285 347 0.285 719 0.286 090 0.286 462 0.286 834 0.287 206 0.287 578 0.287 950 0.288 322 0.288 694 0.289 067 0.289 439 0.289 812 0.290 185 0.290 558 0.290 931 0.291304 0.291 677 0.292 051 0.292 424 0.292 798 0.293 172 0.293 546 0.293 919 0.294 294 369 369 368 369 369 369 369 370 369 370 369 370 370 370 370 370 371 370 371 371 371 370 372 371 371 372 371 372 372 372 372 372 372 372 373 372 373 373 373 373 373 373 374 373 374 374 374 373 375 374 0.204 641 0.205 152 0.205 664 0.206 176 0.206 687 0.207 198 0.207 709 0.208 220 0.208 731 0.209 241 0.209 752 0.210 262 0.210 772 0.211 282 0.211 792 0.212 301 0.212 811 0.213 320 0.213 829 0.214 338 0.214 847 0.215 355 0.215864 0.216 372 0.216 880 0.217 388 0.217 896 0.218 403 0.218 911 0.219 418 0.219 925 0.220 432 0.220 939 0.221 446 0.221 952 0.222 459 0.222 965 0.223 471 0.223 977 0.224 482 0.224 988 0.225 493 0.225 999 0.226 504 0.227 009 0.227 514 0.228 018 0.228 523 0.229 027 0.229 531 511 512 512 511 511 511 511 511 510 511 510 510 510 510 509 510 509 509 509 509 508 509 508 508 508 508 507 508 507 507 507 507 507 506 507 506 506 506 505 506 505 506 505 505 505 504 505 504 504 504 9.928 541 9.928 685 9.928 828 9.928 970 9.929 113 9.929 255 9.929 397 9.929 539 9.929 680 9.929 821 9.929 962 9.930 103 9.930 243 9.930 383 9.930 523 9.930 662 9.930 801 9.930 940 9.931 079 9.931217 9.931 355 9.931 493 9.931 630 9.931 767 9.931 904 9.932 041 9.932 177 9.932 313 9.932 449 9.932 584 9.932 720 9.932 855 9.932 989 9.933 124 9.933 258 9.933 392 9.933 525 9.933 659 9.933 792 9.933 925 9.934 057 9.934 189 9.934 321 9.934 453 9.934 585 9.934 716 9.934 847 9.934 977 9.935 108 9.935 238 11 43 42 43 42 42 42 41 41 41 41 40 40 40 39 39 39 39 88 38 88 87 37 37 87 86 36 36 35 36 36 34 36 34 34 33 34 83 33 32 32 ■■',2 32 32 31 31 30 31 30 30 APPENDIX 311 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS , (Continued) log cosh x log sinh x log tanh i 1.300 1.301 1.302 1.303 1.304 1.305 1.306 1.307 1.308 1.309 1.310 1.311 1.312 1.313 1.314 1.315 1.316 1.317 1.318 1.319 1.320 1.321 1.322 1.323 1.324 1.325 1.326 1.327 1.328 1.329 1.330 1.331 1.332 1.333 1.334 1.335 1.336 1.337 1.338 1.339 1.340 1.341 1.342 1.343 1.344 1.345 1.346 1.347 1.348 1.349 0.294 668 0.295 042 0.295 416 0.295 791 0.296 166 0.296 540 0.296 915 0.297 290 0.297 665 0.298 040 0.298 416 0.298 791 0.299 167 0.299 542 0.299 918 0.300 294 0.300 670 0.301046 0.301422 0.301798 0.302 175 0.302 551 0.302 928 0.303 304 0.303 681 0.304 058 0.304 435 0.304 812 0.305 190 0.305 567 0.305 944 0.306 322 0.306 700 0.307 077 0.307 455 0.307 833 0.308 211 0.308 590 0.308 968 0.309 346 0.309 725 0.310 103 0.310 482 0.310 861 0.311 240 0.311 619 0.311 998 0.312 377 0.312 757 0.313 136 374 374 375 375 374 375 375 375 375 376 375 376 375 376 376 376 376 376 376 377 376 377 376 377 377 377 377 378 377 377 378 378 377 378 378 378 379 378 378 379 378 379 379 379 379 379 379 380 379 380 0.230 035 0.230 539 0.231 043 0.231 547 0.232 050 0.232 554 0.233 057 0.233 560 0.234 063 0.234 565 0.235 068 0.235 570 0.236 073 0.236 575 0.237 077 0.237 579 0.238 080 0.238 582 0.239 083 0.239 584 0.240 086 0.240 586 0.241 087 0.241 588 0.242 089 0.242 589 0.243 089 0.243 589 0.244 089 0.244 589 0.245 089 0.245 588 0.246 088 0.246 587 0.247 086 0.247 585 0.248 084 0.248 583 0.249 081 0.249 580 0.250 0.250 0.251 0.251 0.252 0.252 0.253 0.253 0.254 0.254 078 576 074 572 070 567 065 562 060 557 504 504 504 503 504 503 503 503 502 503 502 503 502 502 502 501 502 501 501 502 500 501 501 501 500 500 500 500 500 500 499 500 499 499 499 499 499 498 499 498 498 498 498 498 497 498 497 498 497 497 9.935 368 9.935 497 9.935 627 9.935 756 9.935 885 9.936 013 9.936 142 9.936 270 9.936 397 9.936 525 9.936 652 9.936 779 9.936 906 9.937 032 9.937 159 9.937 285 9.937 410 9.937 536 9.937 661 9.937 786 9.937 911 9.938 035 9.938 160 9.938 284 9.938 407 9.938 531 9.938 654 9.938 777 9.938 900 9.939 022 9.939 144 9.939 266 9.939 388 9.939 510 9.939 631 9.939 752 9.939 873 9.939 993 9.940 113 9.940 233 9.940 353 9.940 473 9.940 592 9.940 711 9.940 830 9.940 949 9.941067 9.941 185 9.941 303 9.941421 129 130 129 129 128 129 128 127 128 127 127 127 126 127 126 125 126 125 125 125 124 125 124 123 124 123 123 123 122 122 122 122 122 121 121 121 120 120 120 120 120 119 119 119 119 118 118 118 118 117 312 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) !<>K ciwh J log sinh z log tanh x 1.350 1.351 1.352 1.353 1.354 1.355 1.356 1.357 1.358 1.359 1.360 1.361 1.362 1.363 1.364 1.365 1.366 1.367 1.368 1.369 1.370 1.371 1.372 1.373 1.374 1.375 1.376 1.377 1.378 1.379 1.380 1.381 1.382 1.383 1.384 1.385 1.386 1.387 1.388 1.389 1.390 1.391 1.392 1.393 1.394 1.395 1.396 1.397 1.398 1.399 0.313 516 0.313 895 0.314 275 0.314 655 0.315 035 0.315 415 0.315 795 0.316 175 0.316 556 0.316 936 0.317 317 0.317 697 0.318 078 0.318 459 0.318 840 0.319 221 0.319 602 0.319 983 0.320 365 0.320 746 0.321 128 0.321 509 0.321 891 0.322 273 0.322 655 0.323 037 0.323 419 0.323 802 0.324 184 0.324 566 0.324 949 0.325 331 0.325 714 0.326 097 0.326 480 0.326 863 0.327 246 0.327 629 0.328 013 0.328 396 0.32S 0.329 0.329 0.329 0.330 0.330 0.331 0.331 0.331 0.332 780 163 547 931 315 699 083 467 851 235 379 380 380 380 380 380 380 381 380 381 380 381 381 381 381 381 381 382 381 382 381 382 382 382 382 382 383 382 382 383 382 383 383 383 383 383 383 384 383 384 383 384 384 384 384 384 384 384 384 385 0.255 054 0.255 550 0.256 047 0.256 544 0.257 040 0.257 536 0.258 032 0.258 528 0.259 024 0.259 520 0.260 016 0.260 511 0.261006 0.261 502 0.261 997 0.262 492 0.262 987 0.263 481 0.263 976 0.264 470 0.264 965 0.265 459 0.265 953 0.266 447 0.266 941 0.267 434 0.267 928 0.268 421 0.268 915 0.269 408 0.269 901 0.270 394 0.270 886 0.271 379 0.271 872 0.272 364 0.272 856 0.273 349 0.273 841 0.274 333 0.274 824 0.275 316 0.275 808 0.276 299 0.276 790 0.277 282 0.277 773 0.278 264 0.278 754 0.279 245 496 497 497 496 496 496 496 496 496 496 495 495 496 495 495 495 494 495 494 495 494 494 494 494 493 494 493 494 493 493 493 492 493 493 492 492 493 492 492 491 492 492 491 491 492 491 491 490 491 491 9.941 538 9.941 655 9.941 772 9.941 889 9.942 005 9.942 121 9.942 237 0.942 353 9.942 469 9.942 584 9.942 699 9.942 814 9.942 928 9.943 043 9.943 157 9.943 271 9.943 384 9.943 498 9.943 611 9.943 724 9.943 837 9.943 949 9.944 062 9.944 174 9.944 286 9.944 397 9.944 509 9.944 620 9.944 731 9.944 841 9.944 952 9.945 062 9.945 172 9.945 282 9.945 392 9.945 501 9.945 610 9.945 719 9.945 828 9.945 936 9.946 045 9.946 153 9.946 261 9.946 368 9.946 476 9.946 583 9.946 690 9.946 797 9.946 903 9.947 010 17 17 17 If, 10 16 16 16 15 15 15 11 15 14 14 13 14 13 13 13 12 13 12 12 11 12 11 11 10 11 10 10 10 10 09 09 08 09 06 09 OS OS 07 OS 07 07 07 06 07 06 APPENDIX 313 TABLED OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh x log tanh x 1.400 1.401 1.402 1.403 1.404 1.405 1.406 1.407 1.408 1.409 1.410 1.411 1.412 1.413 1.414 1.415 1.416 1.417 1.418 1.419 1.420 1.421 1.422 1.423 1.424 1.425 1.426 1.427 1.428 1.429 1.430 1.431 1.432 433 434 435 436 437 438 439 1.440 1.441 1.442 1.443 1.444 1.445 1.446 1.447 1.448 1.449 0.332 620 0.333 004 0.333 389 0.333 774 0.334 159 0.334 544 0.334 929 0.335 314 0.335 699 0.336 084 C. 336 470 C. 336 855 C. 337 241 C. 337 626 0.338 012 0.338 398 0.338 784 0.339 170 0.339 556 0.339 942 0.340 329 0.340 715 0.341 101 0.341 488 0.341 875 0.342 261 0.342 648 0.343 035 0.343 422 0.343 809 0.344 197 0.344 584 0.344 971 0.345 359 0.345 746 0.346 134 0.346 522 0.346 909 0.347 297 0.347 685 0.348 073 0.348 462 0.348 850 0.349 238 0.349 627 0.350 015 0.350 404 0.350 792 0.351 181 0.351 570 384 385 385 385 385 385 385 385 385 386 385 386 385 386 386 386 386 386 386 387 386 386 387 387 386 387 387 387 387 388 387 387 388 387 388 388 387 388 388 388 389 388 388 389 388 389 388 389 389 389 0.279 736 0.280 226 0.280 717 0.281 207 0.281 697 0.282 187 0.282 677 0.283 167 0.283 656 0.284 146 0.284 635 0.285 124 0.285 614 0.286 103 0.286 592 0.287 080 0.287 569 0.288 058 0.288 546 0.289 035 0.289 523 0.290 011 0.290 499 0.290 987 0.291 475 0.291 962 0.292 450 0.292 937 0.293 425 0.293 912 0.294 399 0.294 886 0.295 373 0.295 860 0.296 346 0.296 833 0.297 319 0.297 806 0.298 292 0.298 778 0.299 264 0.299 750 0.300 236 0.300 721 0.301 207 0.301 692 0.302 178 0.302 663 0.303 148 0.303 633 490 491 490 490 490 490 490 489 490 489 489 490 489 489 488 489 489 488 489 488 488 488 488 488 487 488 487 488 487 487 487 487 487 486 487 486 487 486 486 486 486 486 485 486 485 486 485 485 485 485 9.947 116 9.947 222 9.947 327 9.947 433 9.947 538 9.947 643 9.947 748 9.947 853 9.947 957 9.948 061 9.948 165 9.948 269 9.948 373 9.948 476 9.948 579 9.948 682 9.948 785 9.948 888 9.948 990 9.949 092 9.949 194 9.949 296 9.949 398 9.949 499 9.949 600 9.949 701 9.949 802 9.949 902 9.950 002 9.950 103 9.950 202 9.950 302 9.950 402 9.950 501 9.950 600 9.950 699 9.950 798 9.950 896 9.950 995 9.951093 9.951 191 9.951288 9.951 386 9.951483 9.951580 9.951 677 9.951774 9.951871 9.951 967 9.952 063 106 105 106 105 105 105 105 104 104 104 104 104 103 103 103 103 103 102 102 102 102 102 101 101 101 101 100 100 101 99 100 100 99 99 99 99 98 99 98 98 97 98 97 97 97 97 97 96 96 96 314 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log ooflh z log sinh x log tanh z 1.450 1.451 .452 .453 .454 .455 .456 .457 .458 .459 .460 .461 .462 .463 .464 .465 1.466 1.467 1.468 1.469 1.470 1.471 1.472 1.473 1.474 1.475 1.476 1.477 1.478 1.479 1.480 1.481 1.482 1.483 1.484 1.485 1.486 1.487 1.488 1.489 1.490 1.491 1.492 1.493 1.494 1.495 1.496 1.497 1.498 1.499 0.351959 0.352 348 0.352 737 0.353 126 0.353 516 0.353 905 0.354 295 0.354 684 0.355 074 0.355 463 0.355 853 0.356 243 0.356 633 0.357 023 0.357 413 0.357 804 0.358 194 0.358 584 0.358 975 0.359 365 0.359 756 0.360 147 0.360 537 0.360 928 0.361 319 0.361 710 0.362 101 0.362 493 0.362 884 0.363 275 0.363 667 0.364 058 0.364 450 0.364 842 0.365 233 0.365 625 0.366 017 0.366 409 0.366 801 0.367 194 0.367 586 0.367 978 0.368 371 0.368 763 0.369 156 0.369 548 0.369 941 0.370 334 0.370 727 0.371 120 389 389 389 390 389 390 389 390 389 390 390 390 390 390 391 390 390 391 390 391 391 390 391 391 391 391 392 391 391 392 391 392 392 391 392 392 392 392 393 392 392 393 392 393 392 393 393 393 393 393 0.304 118 0.304 603 0.305 088 0.305 572 0.306 057 0.306 541 0.307 025 0.307 510 0.307 994 0.308 478 0.308 962 0.309 445 0.309 929 0.310 412 0.310 896 0.311 379 0.311 863 0.312 346 0.312 829 0.313 312 0.313 794 0.314 277 0.314 760 0.315 242 0.315 725 0.316 207 0.316 689 0.317 171 0.317 653 0.318 135 0.318 617 0.319 099 0.319 580 0.320 062 0.320 543 0.321 025 0.321 506 0.321 987 0.322 468 0.322 949 0.323 430 0.323 911 0.324 391 0.324 872 0.325 352 0.325 832 0.326 313 0.326 793 0.327 273 0.327 753 485 485 484 485 484 484 485 484 484 484 483 484 483 484 483 484 483 483 483 482 483 483 482 483 482 482 482 482 482 482 482 481 482 481 482 481 481 481 481 481 481 480 481 480 480 481 480 480 480 480 9.952 159 9.952 255 9.952 350 9.952 446 9.952 541 9.952 636 9.952 731 9.952 826 9.952 920 9.953 014 9.953 108 9.953 202 9.953 296 9.953 389 9.953 483 9.953 576 9.953 669 9.953 761 9.953 854 9.953 946 9.954 039 9.954 131 9.954 222 9.954 314 9.954 405 9.954 497 9.954 588 9.954 679 9.954 769 9.954 860 9.954 950 9.955 040 9.955 130 9.955 220 9.955 310 9.955 399 9.955 489 9.955 578 9.955 667 9.955 755 9.955 844 9.955 932 9.956 020 9.956 109 9.956 196 9.956 284 9.956 372 9.956 459 9.956 546 9.956 633 96 95 96 95 95 95 95 94 94 94 94 94 93 94 93 93 92 93 92 93 92 91 92 91 92 91 91 90 91 90 90 90 90 90 89 90 89 89 88 89 88 88 89 87 88 88 87 87 87 87 APPENDIX 315 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) 1.500 1.501 1.502 1.503 1.504 1.505 1.506 1.507 1.508 1.509 1.510 1.511 1.512 1.513 1.514 1.515 1.516 1.517 1.518 1.519 1.520 1.521 1.522 1.523 1.524 1.525 1.526 1.527 1.528 1.529 1.530 1.531 1.532 1.533 1.534 1.535 1.536 1.537 1.538 1.539 1.540 1.541 1.542 1.543 1.544 1.545 1.546 1.547 1.548 1.549 log cosh x 0.371513 0.371906 0.372 299 0.372 693 0.373 086 0.373 479 0.373 873 0.374 267 0.374 660 0.375 054 0.375 448 0.375 842 0.376 236 0.376 630 0.377 024 0.377 418 0.377 813 0.378 207 0.378 601 0.378 996 0.379 390 0.379 785 0.380 180 0.380 575 0.380 970 0.381365 0.381 760 0.382 155 0.382 550 0.382 945 0.383 341 0.383 736 0.384 132 0.384 527 0.384 923 0.385 319 0.385 714 0.386 110 0.386 506 0.386 902 0.387 298 0.387 694 0.388 091 0.388 487 0.388 883 0.389 280 0.389 676 0.390 073 0.390 470 0.390 866 393 393 394 393 393 394 394 393 394 394 394 394 394 394 394 395 394 394 395 394 395 395 395 395 395 395 395 395 395 396 395 396 395 396 396 395 396 396 396 396 396 397 396 396 397 396 397 397 396 397 log sinh x 0.328 233 0.328 712 0.329 192 0.329 672 0.330 151 0.330 630 0.331 110 0.331 589 0.332 068 0.332 547 0.333 026 0.333 505 0.333 983 0.334 462 0.334 941 0.335 419 0.335 897 0.336 376 0.336 854 0.337 332 0.337 810 0.338 288 0.338 766 0.339 243 0.339 721 0.340 198 0.340 676 0.341 153 0.341 630 0.342 108 0.342 585 0.343 062 0.343 539 0.344 015 0.344 492 0.344 969 0.345 445 0.345 922 0.346 398 0.346 874 0.347 350 0.347 827 0.348 303 0.348 778 0.349 254 0.349 730 0.350 206 0.350 681 0.351 157 0.351 632 479 480 480 479 479 480 479 479 479 479 479 478 479 479 478 478 479 478 478 478 478 478 477 478 477 478 477 477 478 477 477 477 476 477 477 476 477 476 476 476 477 476 475 476 476 476 475 476 475 475 log tanh z 9.956 720 9.956 806 9.956 893 9.956 979 9.957 065 9.957 151 9.957 237 9.957 322 9.957 408 9.957 493 9.957 578 9.957 663 9.957 748 9.957 832 9.957 917 9.958 001 9.958 085 9.958 169 9.958 252 9.958 336 9.958 419 9.958 503 9.958 586 9.958 669 9.958 751 9.958 834 9.958 916 9.958 998 9.959 080 9.959 162 9.959 244 9.959 326 9.959 407 9.959 488 9.959 569 9.959 650 9.959 731 9.959 811 9.959 892 9.959 972 9.960 052 9.960 132 9.960 212 9.960 291 9.960 371 9.960 450 9.960 529 9.960 608 9.960 687 9.960 766 87 86 86 85 86 85 85 85 85 84 85 84 84 84 83 84 83 84 83 83 82 83 82 82 82 82 82 82 81 81 81 81 81 80 81 80 80 80 80 79 80 79 79 79 79 79 78 316 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh z log sinh x 1(>k tanli z 1.550 1.551 1.552 1.553 1.554 1.555 1.556 1.557 1.558 1.559 1.560 1.561 1.562 1.563 1.564 1.565 1.566 1.567 1.568 1.569 570 571 572 573 574 1.575 1.576 1.577 1.578 1.579 1.580 1.581 1.582 1.583 1.584 1.585 1.586 1.587 1.588 1.589 1.590 1.591 1.592 1.593 1.594 1.595 1.596 1.597 1.598 1.599 0.391 263 0.391 660 0.392 057 0.392 454 0.392 851 0.393 248 0.393 646 0.394 043 0.394 440 0.394 838 0.395 235 0.395 633 0.396 031 0.396 428 0.396 826 0.397 224 0.397 622 0.398 020 0.398 418 0.398 816 0.399 214 0.399 613 0.400 011 0.400 409 0.400 808 0.401 207 0.401 605 0.402 004 0.402 403 0.402 802 0.403 200 0.403 599 0.403 998 0.404 398 0.404 797 0.405 196 0.405 595 0.405 995 0.406 394 0.406 794 0.407 193 0.407 593 0.407 993 0.408 392 0.408 792 0.409 192 0.409 592 0.409 992 0.410 392 0.410 792 397 397 397 397 397 398 397 397 398 397 398 398 397 398 398 398 398 398 398 398 399 398 398 399 399 398 399 399 399 398 399 399 400 399 399 399 400 399 400 399 400 400 399 400 400 400 400 400 400 401 0.352 107 0.352 583 0.353 058 0.353 533 0.354 008 0.354 483 0.354 958 0.355 432 0.355 907 0.356 381 0.356 856 0.357 330 0.357 805 0.358 279 0.358 753 0.359 227 0.359 701 0.360 175 0.360 649 0.361 122 0.361 596 0.362 069 0.362 543 0.363 016 0.363 490 0.363 963 0.364 436 0.364 909 0.365 382 0.365 855 0.366 328 0.366 800 0.367 273 0.367 746 0.368 218 0.368 691 0.369 163 0.369 635 0.370 107 0.370 580 0.371 052 0.371 523 0.371 995 0.372 467 0.372 939 0.373 410 0.373 882 0.374 353 0.374 825 0.375 296 476 475 475 475 475 475 474 475 474 475 474 475 474 474 474 474 474 474 473 474 473 474 473 474 473 473 473 473 473 473 472 473 473 472 473 472 472 472 473 472 471 472 472 472 471 472 471 472 471 471 9.960 844 9.960 923 9.961001 9.961079 9.961 157 9.961234 9.961312 9.961 389 9.961 467 9.961544 9.961621 9.961697 9.961774 9.961850 9.961 927 9.962 003 9.962 079 9.962 155 9.962 231 9.962 306 9.962 382 9.962 457 9.962 532 9.962 607 9.962 682 9.962 756 9.962 831 9.962 905 9.962 979 9.963 053 9.963 127 9.963 201 9.963 275 9.963 348 9.963 421 9.963 495 9.963 568 9.963 641 9.963 713 9.963 786 9.963 858 9.963 931 9.964 003 9.964 075 9.964 147 9.964 218 9.964 290 9.964 361 9.964 433 9.964 504 APPENDIX 317 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh x log sinh z log tanh x 1.600 1.601 1.602 1.603 1.604 1.605 1.606 1.607 1.608 1.609 1.610 1.611 1.612 1.613 1.614 1.615 1.616 1.617 1.618 1.619 1.620 .621 .622 .623 .624 .625 1.626 1.627 1.628 1.629 1.630 1.631 1.632 1.633 1.634 1.635 1.636 1.637 1.638 1.639 1.640 1.641 1.642 1.643 1.644 1.645 1.646 1.647 1.648 1.649 0.411 193 0.411 593 0.411 993 0.412 394 0.412 794 0.413 195 0.413 596 0.413 996 0.414 397 0.414 798 0.415 199 0.415 600 0.416 001 0.416 402 0.416 803 0.417 204 0.417 605 0.418 007 0.418 408 0.418 810 0.419 211 0.419 613 0.420 014 0.420 416 0.420 818 0.421 220 0.421 622 0.422 024 0.422 426 0.422 828 0.423 230 0.423 632 0.424 034 0.424 437 0.424 839 0.425 242 0.425 644 0.426 047 0.426 449 0.426 852 0.427 255 0.427 658 0.428 060 0.428 463 0.428 866 0.429 269 0.429 673 0.430 076 0.430 479 0.430 882 400 400 401 400 401 401 400 401 401 401 401 401 401 401 401 401 402 401 402 401 402 401 402 402 402 402 402 402 402 402 402 402 403 402 403 402 403 402 403 403 403 402 403 403 403 404 403 403 403 404 0.375 767 0.376 239 0.376 710 0.377 181 0.377 652 0.378 123 0.378 593 0.379 064 0.379 535 0.380 005 0.380 476 0.380 946 0.381 416 0.381 887 0.382 357 0.382 827 0.383 297 0.383 767 0.384 237 0.384 707 0.385 176 0.385 646 0.386 116 0.386 585 0.387 055 0.387 524 0.387 993 0.388 462 0.388 932 0.389 401 0.389 870 0.390 339 0.390 807 0.391 276 0.391 745 0.392 214 0.392 682 0.393 151 0.393 619 0.394 088 0.394 556 0.395 024 0.395 492 0.395 960 0.396 428 0.396 896 0.397 364 0.397 832 0.398 300 0.398 767 472 471 471 471 471 470 471 471 470 471 470 470 471 470 470 470 470 470 470 469 470 470 469 470 469 469 469 470 469 469 469 468 469 469 469 468 469 468 469 468 468 468 468 468 468 468 468 468 467 468 9.964 575 9.964 646 9.964 716 9.964 787 9.964 857 9.964 928 9.964 998 9.965 068 9.965 138 9.965 207 9.965 277 9.965 346 9.965 416 9.965 485 9.965 554 9.965 623 9.965 692 9.965 760 9.965 829 9.965 897 9.965 965 9.966 033 9.966 101 9.966 169 9.966 237 9.966 304 9.966 372 9.966 439 9.966 506 9.966 573 9.966 640 9.966 707 9.966 773 9.966 840 9.966 906 9.966 972 9.967 038 9.967 104 9.967 170 9.967 236 9.967 301 9.967 366 9.967 432 9.967 497 9.967 562 9.967 627 9.967 691 9.967 756 9.967 821 9.967 885 71 70 71 70 71 70 70 70 69 70 69 70 69 69 69 69 68 69 68 68 6? 68 68 68 67 68 67 67 67 67 67 66 67 66 66 66 66 66 66 65 65 66 65 65 65 64 65 65 64 64 318 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) 1.650 1.651 1.652 1.653 1.654 1.655 1.656 1.657 1.658 1.659 1.660 1.661 1.662 1.663 1.664 1.665 1.666 1.667 1.668 1.669 1.670 1.671 1.672 1.673 1.674 1.675 1.676 1.677 1.678 1.679 1.680 1.681 1.682 1.683 1.684 1.685 1.686 1.687 1.688 1.689 1.690 1.691 1.692 1.693 1.694 1.695 1.696 1.697 1.698 1.699 1«>C n.-h i 0.431286 0.431 689 0.432 093 0.432 496 0.432 900 0.433 303 0.433 707 0.434 111 0.434 515 0.434 919 0.435 323 0.435 727 0.436 131 0.436 535 0.436 939 0.437 343 0.437 748 0.438 152 0.438 556 0.438 961 0.439 365 0.439 770 0.440 175 0.440 579 0.440 984 0.441 389 0.441 794 0.442 199 0.442 604 0.443 009 0.443 414 0.443 819 0.444 224 0.444 630 0.445 035 0.445 440 0.445 846 0.446 251 0.446 657 0.447 062 0.447 468 0.447 874 0.448 280 0.448 685 0.449 091 0.449 497 0.449 903 0.450 309 0.450 715 0.451 122 403 404 403 404 403 404 404 404 404 404 404 404 404 404 404 405 404 404 405 404 405 405 404 405 405 405 405 405 405 405 405 405 406 405 405 406 405 406 405 406 406 406 405 406 406 406 406 406 407 406 log sinh z 0.399 235 0.399 702 0.400 170 0.400 637 0.401 105 0.401 572 0.402 039 0.402 506 0.402 973 0.403 440 0.403 907 0.404 374 0.404 841 0.405 307 0.405 774 0.406 241 0.406 707 0.407 173 0.407 640 0.408 106 0.408 572 0.409 039 0.409 505 0.409 971 0.410 437 0.410 903 0.411 368 0.411 834 0.412 300 0.412 766 0.413 231 0.413 697 0.414 162 0.414 628 0.415 093 0.415 558 0.416 023 0.416 488 0.416 954 0.417 419 0.417 883 0.418 348 0.418 813 0.419 278 0.419 743 0.420 207 0.420 672 0.421 136 0.421 601 0.422 065 467 468 467 468 467 467 467 467 467 467 467 467 466 467 467 466 466 467 466 466 467 466 466 466 466 465 466 466 466 465 466 465 466 465 465 465 465 466 465 464 465 465 465 465 464 465 464 465 464 464 log tanli i 9.967 949 9.968 013 9.968 077 9.968 141 9.968 205 9.968 268 9.968 332 9.968 395 9.968 458 9.968 521 9.968 584 9.968 647 9.968 710 9.968 772 9.968 835 9.968 897 9.968 959 9.969 021 9.969 083 9.969 145 9.969 207 9.969 269 9.969 330 9.969 391 9.969 453 9.969 514 9.969 575 9.969 635 9.969 696 9.969 757 9.969 817 9.969 878 9.969 938 9.969 998 9.970 058 9.970 118 9.970 178 9.970 237 9.970 297 9.970 356 9.970 415 9.970 474 9.970 534 9.970 592 9.970 651 9.970 710 9.970 768 9.970 827 9.970 885 9.970 943 64 64 64 64 63 64 63 63 63 63 63 63 62 63 62 62 62 62 62 62 62 61 61 62 61 61 60 61 61 60 61 60 60 60 60 60 59 '60 59 59 59 60 58 59 59 58 59 58 58 59 APPENDIX 319 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh z log sinh / log tanh i 1.700 1.701 1.702 1.703 1.704 1.705 1.706 1.707 1.708 1.709 1.710 1.711 1.712 1.713 1.714 1.715 1.716 1.717 1.718 1.719 1.720 1.721 1.722 1.723 1.724 1.725 1.726 1.727 1.728 1.729 1.730 1.731 1.732 1.733 1.734 1.735 1.736 1.737 1.738 1.739 1.740 1.741 1.742 1.743 1.744 1.745 1.746 1.747 1.748 1.749 0.451 528 0.451934 0.452 340 0.452 747 0.453 153 0.453 560 0.453 966 0.454 373 0.454 780 0.455 186 0.455 593 0.456 000 0.456 407 0.456 814 0.457 221 0.457 628 0.458 035 0.458 442 0.458 849 0.459 256 0.459 663 0.460 071 0.460 478 0.460 886 0.461 293 0.461 701 0.462 108 0.462 516 0.462 924 0.463 331 0.463 739 0.464 147 0.464 555 0.464 963 0.465 371 0.465 779 0.466 187 0.466 595 0.467 003 0.467 412 0.467 820 0.468 228 0.468 637 0.469 045 0.469 454 0.469 862 0.470 271 0.470 680 0.471 088 0.471 497 406 406 407 406 407 406 407 407 406 407 407 407 407 407 407 407 407 407 407 407 408 407 408 407 408 407 408 408 407 408 408 408 408 408 408 408 408 408 409 408 408 409 408 409 408 409 409 408 409 409 0.422 529 0.422 994 0.423 458 0.423 922 0.424 386 0.424 850 0.425 314 0.425 778 0.426 242 0.426 705 0.427 169 0.427 633 0.428 096 0.428 560 0.429 023 0.429 487 0.429 950 0.430 413 0.430 877 0.431 340 0.431 803 0.432 266 0.432 729 0.433 192 0.433 655 0.434 117 0.434 580 0.435 043 0.435 506 0.435 968 0.436 431 0.436 893 0.437 355 0.437 818 0.438 280 0.438 742 0.439 204 0.439 667 0.440 129 0.440 591 0.441052 0.441514 0.441976 0.442 438 0.442 900 0.443 361 0.443 823 0.444 284 0.444 746 0.445 207 465 464 464 464 464 464 464 464 463 464 464 463 464 463 464 463 463 464 463 463 463 463 463 463 462 463 463 463 462 463 462 462 463 462 462 462 463 462 462 461 462 462 462 462 461 462 461 462 461 462 9.971 002 9.971060 9.971 117 9.971 175 9.971233 9.971 290 9.971 348 9.971405 9.971 462 9.971519 9.971 576 9.971633 9.971 690 9.971 746 9.971803 9.971 859 9.971915 9.971 972 9.972 028 9.972 084 9.972 139 9.972 195 9.972 251 9.972 306 9.972 362 9.972 417 9.972 472 9.972 527 9.972 582 9.972 637 9.972 691 9.972 746 9.972 801 9.972 855 9.972 909 9.972 963 9.973 017 9.973 071 9.973 125 9.973 179 9.973 233 9.973 286 9.973 339 9.973 393 9.973 446 9.973 499 9.973 552 9.973 605 9.973 658 9.973 710 58 57 58 58 57 58 57 57 57 57 57 57 56 57 56 56 57 56 56 55 56 56 55 56 55 55 55 55 55 54 55 55 54 54 54 54 54 54 54 54 53 '53 54 53 53 53 53 53 52 53 320 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) 1<>K cosh i log sinh z log tanh z 1.750 0.471 906 1.751 0.472 315 1.752 0.472 724 1.753 0.473 133 1.754 0.473 542 1.755 0.473 951 1.756 0.474 360 1.757 0.474 769 1.758 0.475 178 1.759 0.475 587 1.760 0.475 997 1.761 0.476 406 1.762 0.476 815 1.763 0.477 225 1.764 0.477 634 1.765 0.478 044 1.766 0.478 453 1.767 0.478 863 1.768 0.479 273 1.769 0.479 682 1.770 0.480 092 1.771 0.480 502 1.772 0.480 912 1.773 0.481322 1.774 0.481 732 1.775 0.482 142 1.776 0.482 552 1.777 0.482 962 1.778 0.483 372 1.779 0.483 782 1.780 0.484 193 1.781 0.484 603 1.782 0.485 013 1.783 0.485 424 1.784 0.485 834 1.785 0.486 245 1.786 0.486 655 1.787 0.487 066 1.788 0.487 476 1.789 0.487 887 1.790 0.488 298 1.791 0.488 708 1.792 0.489 119 1.793 0.489 530 1.794 0.489 941 1.795 0.490 352 1.796 0.490 763 1.797 0.491 174 1.798 0.491 585 1.799 0.491 996 409 409 409 409 409 409 409 409 409 410 409 409 410 409 410 409 410 410 409 410 410 410 410 410 410 410 410 410 410 411 410 410 411 410 411 410' 411 410 411 411 410 411 411 411 411 411 411 411 411 411 0.445 669 0.446 130 0.446 591 0.447 052 0.447 514 0.447 975 0.448 436 0.448 897 0.449 358 0.449 819 0.450 279 0.450 740 0.451 201 0.451661 0.452 122 0.452 583 0.453 043 0.453 504 0.453 964 0.454 424 0.454 885 0.455 345 0.455 805 0.456 265 0.456 725 0.457 185 0.457 645 0.458 105 0.458 565 0.459 025 0.459 484 0.459 944 0.460 404 0.460 863 0.461 323 0.461 782 0.462 242 0.462 701 0.463 160 0.463 620 0.464 079 0.464 538 0.464 997 0.465 456 0.465 915 0.466 374 0.466 833 0.467 292 0.467 751 0.468 210 461 461 461 462 461 461 461 461 461 460 461 461 460 461 461 460 461 460 460 461 460 460 .460 460 460 460 460 460 460 459 460 460 459 460 459 460 459 459 460 459 459 459 459 459 459 459 459 459 459 458 9.973 763 9.973 815 9.973 868 9.973 920 9.973 972 9.974 024 9.974 076 9.974 128 9.974 180 9.974 231 9.974 283 9.974 334 9.974 385 9.974 437 9.974 488 9.974 539 9.974 590 9.974 640 9.974 691 9.974 742 9.974 792 9.974 843 9.974 893 9.974 943 9.974 993 9.975 043 9.975 093 9.975 143 9.975 193 9.975 242 9.975 292 9.975 341 9.975 390 9.975 440 9.975 489 9.975 538 9.975 587 9.975 635 9.975 684 9.975 733 9.975 781 9.975 830 9.975 878 9.975 926 9.975 974 9.976 022 9.976 070 9.976 118 9.976 166 9.976 213 52 53 52 52 52 52 52 52 51 52 51 51 52 51 51 51 50 51 51 50 51 50 50 50 50 50 50 50 49 50 49 49 50 49 49 49 48 49 49 48 49 48 48 48 48 48 48 48 47 48 APPENDIX 321 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh z log si nil X log tanh x 1.800 1.801 1.802 1.803 1.804 1.805 1.806 1.807 1.808 1.809 1.810 1.811 1.812 1.813 1.814 1.815 1.816 1.817 1.818 1.819 1.820 1.821 1.822 1.823 1.824 1.825 1.826 1.827 1.828 1.829 1.830 1.831 1.832 1.833 1.834 1.835 1.836 1.837 1.838 1.839 1.840 1.841 .842 .843 .844 .845 .846 .847 1.848 1.849 0.492 407 0.492 819 0.493 230 0.493 641 0.494 053 0.494 464 0.494 875 0.495 287 0.495 698 0.496 110 0.496 522 0.496 933 0.497 345 0.497 757 0.498 168 0.498 580 0.498 992 0.499 404 0.499 816 0.500 228 0.500 640 0.501 052 0.501 464 0.501 877 0.502 289 0.502 701 0.503 113 0.503 526 0.503 938 0.504 351 0.504 763 0.505 176 0.505 588 0.506 001 0.506 413 0.506 826 0.507 239 0.507 652 0.508 064 0.508 477 0.508 890 0.509 303 0.509 716 0.510 129 0.510 542 0.510 955 0.511 368 0.511 782 0.512 195 0.512 608 412 411 411 412 411 411 412 411 412 412 411 412 412 411 412 412 412 412 412 412 412 412 413 412 412 412 413 412 413 412 413 412 413 412 413 413 413 412 413 413 413 413 413 413 413 413 414 413 413 414 0.468 668 0.469 127 0.469 586 0.470 044 0.470 503 0.470 961 0.471 420 0.471 878 0.472 336 0.472 795 0.473 253 0.473 711 0.474 169 0.474 627 0.475 085 0.475 543 0.476 001 0.476 459 0.476 917 0.477 375 0.477 832 0.478 290 0.478 748 0.479 205 0.479 663 0.480 120 0.480 578 0.481 035 0.481 493 0.481 950 0.482 407 0.482 864 0.483 321 0.483 779 0.484 236 0.484 693 0.485 150 0.485 607 0.486 064 0.486 520 0.486 977 0.487 434 0.487 891 0.488 347 0.488 804 0.489 260 0.489 717 0.490 173 0.490 630 0.491 086 459 459 458 459 458 459 458 458 459 458 458 458 458 458 458 458 458 458 458 457 458 458 457 458 457 458 457 458 457 457 457 457 458 457 457 457 457 457 456 457 457 457 456 457 456 457 456 457 456 457 9.976 261 9.976 308 9.976 356 9.976 403 9.976 450 9.976 497 9.976 544 9.976 591 9.976 638 9.976 685 9.976 731 9.976 778 9.976 824 9.976 871 9.976 917 9.976 963 9.977 009 9.977 055 9.977 101 9.977 147 9.977 192 9.977 238 9.977 283 9.977 329 9.977 374 9.977 419 9.977 464 9.977 509 9.977 554 9.977 599 9.977 644 9.977 689 9.977 733 9.977 778 9.977 822 9.977 867 9.977 911 9.977 955 9.977 999 9.978 043 9.978 087 9.978 131 9.978 174 9.978 218 9.978 262 9.978 305 9.978 348 9.978 392 9.978 435 9.978 478 47 48 47 47 47 47 47 47 47 46 47 46 47 46 46 46 46 46 46 45 46 45 46 45 45 45 45 45 45 45 45 44 45 44 45 44 44 44 44 44 44 43 44 44 43 43 44 43 43 43 322 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Continued) log cosh z log hi ii h z log t null z 1.850 1.851 1.S52 1.853 1.854 1.855 1.856 1.857 1.858 1.859 1.860 1.861 1.862 1.863 1.864 1.865 1.866 1.867 1.868 1.869 1.870 1.871 1.872 873 ,874 .875 .876 .877 .878 .879 1.880 1.881 1.882 1.883 1.884 1.885 1.886 1.887 1.888 1.889 1.890 1.891 1.892 1.893 1.894 1.895 1.896 1.897 1.898 1.899 0.513 022 0.513 435 0.513 848 0.514 262 0.514 675 0.515 089 0.515 502 0.515 916 0.516 330 0.516 743 0.517 157 0.517 571 0.517 984 0.518 398 0.518 812 0.519 226 0.519 640 0.520 054 0.520 468 0.520 882 0.521 296 0.521 711 0.522 125 0.522 539 0.522 953 0.523 368 0.523 782 0.524 196 0.524 611 0.525 025 0.525 440 0.525 854 0.526 269 0.526 683 0.527 098 0.527 513 0.527 928 0.528 342 0.528 757 0.529 172 0.529 587 0.530 002 0.530 417 0.530 832 0.531 247 0.531 662 0.532 077 0.532 492 0.532 907 0.533 323 413 413 414 413 414 413 414 414 413 414 414 413 414 414 414 414 414 414 414 414 415 414 414 414 415 414 414 415 414 415 414 415 414 415 415 415 414 415 415 415 415 415 415 415 415 415 415 415 416 415 0.491 543 0.491 999 0.492 455 0.492 911 0.493 367 0.493 824 0.494 280 0.494 736 0.495 192 0.495 648 0.496103 0.496 559 0.497 015 0.497 471 0.497 927 0.498 382 0.498 838 0.499 293 0.499 749 0.500 204 0.500 660 0.501 115 0.501571 0.502 026 0.502 481 0.502 937 0.503 392 0.503 847 0.504 302 0.504 757 0.505 212 0.505 667 0.506 122 0.506 577 0.507 032 0.507 487 0.507 941 0.508 396 0.508 851 0.509 305 0.509 760 0.510 215 0.510 669 0.511 124 0.511 578 0.512 032 0.512 487 0.512 941 0.513 395 0.513 850 456 456 456 456 457 456 456 456 456 455 456 456 456 456 455 456 455 456 455 456 455 456 455 455 456 455 455 455 455 455 455 455 455 455 455 454 455 455 454 455 455 454 455 454 454 455 454 454 455 454 9.978 521 9.978 564 9.978 607 9.978 650 9.978 692 9.978 735 9.978 777 9.978 820 9.978 862 9.978 904 9.978 947 9.978 989 9.979 031 9.979 073 9.979 114 9.979 156 9.979 198 9.979 239 9.979 281 9.979 322 9.979 364 9.979 405 9.979 446 9.979 487 9.979 528 9.979 569 9.979 610 9.979 651 9.979 691 9.979 732 9.979 772 9.979 813 9.979 853 9.979 893 9.979 934 9.979 974 9.980 014 9.980 054 9.980 094 9.980 133 9.980 173 9.980 213 9.980 252 9.980 292 9.980 331 9.980 370 9.980 410 9.980 449 9.980 488 9.880 527 APPENDIX 323 TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS {Continued) log cosh x log sinh x log tanh x 1.900 1.901 1.902 1.903 1.904 1.905 1.906 1.907 1.908 1.909 1.910 1.911 1.912 1.913 1.914 1.915 1.916 1.917 1.918 1.919 1.920 1.921 1.922 1.923 1.924 1.925 1.926 1.927 1.928 1.929 1.930 1.931 1.932 1.933 1.934 1.935 1.936 1.937 1.938 1.939 1.940 1.941 1.942 1.943 1.944 1.945 1.946 1.947 1.948 1.949 0.533 738 0.534 153 0.534 569 0.534 984 0.535 399 0.535 815 0.536 230 0.536 646 0.537 061 0.537 477 0.537 893 0.538 308 0.538 724 0.539 140 0.539 556 0.539 971 0.540 387 0.540 803 0.541 219 0.541 635 0.542 051 0.542 467 0.542 883 0.543 299 0.543 715 0.544 132 0.544 548 0.544 964 0.545 380 0.545 797 0.546 213 0.546 629 0.547 046 0.547 462 0.547 879 0.548 295 0.548 712 0.549 128 0.549 545 0.549 962 0.550 378 0.550 795 0.551 212 0.551 629 0.552 046 0.552 463 0.552 879 0.553 296 0.553 713 0.554 130 415 416 415 415 416 415 416 415 416 416 415 416 416 416 415 416 416 416 416 416 416 416 416 416 417 416 416 416 417 416 416 417 416 417 416 417 416 417 417 416 417 417 417 417 417 416 417 417 417 417 0.514 304 0.514 758 0.515 212 0.515 666 0.516 120 0.516 574 0.517 028 0.517 482 0.517 936 0.518 390 0.518 843 0.519 297 0.519 751 0.520 205 0.520 658 0.521 112 0.521565 0.522 019 0.522 472 0.522 926 0.523 379 0.523 832 0.524 286 0.524 739 0.525 192 0.525 645 0.526 099 0.526 552 0.527 005 0.527 458 0.527 911 0.528 364 0.528 817 0.529 270 0.529 723 0.530 175 0.530 628 0.531081 0.531 534 0.531 986 0.532 439 0.532 891 0.533 344 0.533 797 0.534 249 0.0534 71 0.535 154 0.535 606 0.536 059 0.536 511 454 454 454 454 454 454 454 454 454 453 454 454 454 453 454 453 454 453 454 453 453 454 453 453 453 454 453 453 453 453 453 453 453 453 452 453 453 453 452 453 452 453 453 452 452 453 452 453 452 452 9.980 566 9.980 605 9.980 643 9.980 682 9.980 721 9.980 759 9.980 798 9.980 836 9.980 874 9.980 913 9.980 951 9.980 989 9.981 027 9.981065 9.981 103 9.981 140 9.981 178 9.981 216 9.981253 9.981 291 9.981328 9.981 365 9.981 403 9.981 440 9.981 477 9.981 514 9.981 551 9.981 588 9.981 624 9.981 661 9.981698 9.981 734 9.981 771 9.981 807 9.981 844 9.981 880 9.981 916 9.981 952 9.981988 9.982 024 9.982 060 9.982 096 9.982 132 9.982 168 9.982 203 9.982 239 9.982 274 9.982 310 9.982 345 9.982 380 39 38 39 39 38 39 38 38 39 38 38 38 38 38 37 38 38 37 38 37 37 38 37 37 37 37 37 37 37 37 36 37 36 37 36 36 36 36 36 36 36 36 36 35 36 35 36 35 35 36 324 APPENDIX TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS (Concluded) log cosh z log sinh x log tanh x 1.950 1.951 1.952 1.953 1.954 1.955 1.956 1.957 1.958 1.959 1.960 1.961 1.962 1.963 1.964 1.965 1.966 1.967 1.968 1.969 1.970 1.971 1.972 1.973 1.974 1.975 1.976 1.977 1.978 1.979 1.980 1.981 1.982 1.983 1.984 1.985 1.986 1.987 1.988 1.989 1.990 1.991 1.992 1.993 1.994 1.995 1.996 1.997 1.998 1.999 2.000 0.554 547 0.554 964 0.555 382 0.555 799 0.556 216 0.556 633 0.557 050 0.557 468 0.557 885 0.558 302 0.558 720 0.559 137 0.559 555 0.559 972 0.560 390 0.560 807 0.561 225 0.561 642 0.562 060 0.562 478 0.562 895 0.563 313 0.563 731 0.564 149 0.564 566 0.564 984 0.565 402 0.565 820 0.566 238 0.566 656 0.567 074 0.567 492 0.567 910 0.568 328 0.568 747 0.569 165 0.569 583 0.570 001 0.570 420 0.570 838 0.571 256 0.571 675 0.572 093 0.572 511 0.572 930 0.573 348 0.573 767 0.574 186 0.574 604 0.575 023 0.575 441 417 418 417 417 417 417 418 417 417 418 417 418 417 418 417 418 417 418 418 417 418 418 418 417 418 418 418 418 418 418 418 418 418 419 418 418 418 419 418 418 419 418 418 419 418 419 419 418 419 418 0.536 963 0.537 415 0.537 868 0.538 320 0.538 772 0.539 224 0.539 676 0.540 128 0.540 580 0.541 032 0.541 484 0.541 936 0.542 387 0.542 839 0.543 291 0.543 743 0.544 194 0.544 646 0.545 098 0.545 549 0.546 001 0.546 452 0.546 904 0.547 355 0.547 806 0.548 258 0.548 709 0.549 160 0.549 612 0.550 063 0.550 514 0.550 965 0.551 416 0.551 868 0.552 319 0.552 770 -0.553 221 0.553 672 0.554 122 0.554 573 0.555 024 0.555 475 0.555 926 0.556 377 0.556 827 0.557 278 0.557 729 0.558 179 0.558 630 0.559 080 0.559 531 452 453 452 452 452 452 452 452 452 452 452 451 452 452 452 451 452 452 451 452 451 452 451 451 452 451 451 452 451 451 451 451 452 451 451 451 451 450 451 451 451 451 451 450 451 451 450 451 450 451 9.982 416 9.982 451 9.982 486 9.982 521 9.982 556 9.982 591 9.982 626 9.982 660 9.982 695 9.982 729 9.982 764 9.982 798 9.982 833 9.982 867 9.982 901 9.982 936 9.982 970 9.983 004 9.983 038 9.983 072 9.983 105 9.983 139 9.983 173 9.983 206 9.983 240 9.983 274 9.983 307 9.983 340 9.983 374 9.983 407 9.983 440 9.983 473 9.983 506 9.983 539 9.983 572 9.983 605 9.983 638 9.983 670 9.983 703 9.983 735 9.983 768 9.983 800 9.983 833 9.983 865 9.983 897 9.983 930 9.983 962 9.983 994 9.984 026 9.984 058 9.984 089 35 35 35 35 35 35 34 35 34 35 34 35 34 34 35 34 34 34 34 33 34 34 33 34 34 33 33 34 33 33 33 33 33 33 33 33 32 33 32 33 32 33 32 32 33 32 32 32 32 31 APPENDIX 325 Curve No. FORM A Separation of Odd and Even Harmonics for 72 Ordinate Analysis Vm' — 5 (Vm — ym+ii). y» h (ym + y m +3t,). Data Diff. Sum. Data Diff. Sum. tn Vm Vm 2Vm' 2»„" Vm Vm 2Vm 2Vm" 36 18 54 l 37 19 55 2 38 20 56 3 39 21 57 4 40 22 58 5 41 23 59 6 42 24 60 7 43 25 61 8 44 26 62 9 45 27 63 10 46 28 64 11 47 29 65 12 48 30 66 13 49 31 67 14 50 32 68 15 51 33 69 16 52 34 70 17 53 35 71 Use y m ' in analysis for odd harmonics. Carry y m " to Form E as data. Remarks. Computed by 326 AIM'KNDIX Curve No. 95 .03 R 0* « 05 c O — OrtfCCI^OCOO — N«*iO«NOO«O^NW , *tOONOOfflp-^«M<»'IO« - n 11 11 11 n a n n 1 1 1 1 1 1 1 11 • 1 1 . . . CQ02E003&3 «3 QQQQQQQQ - - • - ►• !T»~« L - ', • • +++++ 1 + 1 QQQQQQQQ 0305 oj crj 03 03 zqcq b 1 » 11 11 11 11 n 1 1 ■ 1 ■ g ■ 1 QQQQQQQQ 1303 go 0303 cocce? £ 00 ■ 0Q s e ^^cin^tiocDt-QQoto^nec^iOttNoc 6= 5 1 t cceoc«5eococ«5e<ic*eie*c*c^c*e*«ca-* s 1 Q 1 1 p e >««<n4'ioot<oooio>'Neo«ice^i> : «5K g APPENDIX 327 ■OR aAHHQ 03 I o wjsp uiojj ajBuipio pajnduioo ajwutpjo • S O!DON0t)TfNO a '5 o '3 3 d .2 1 8 o .§ O «5 ■ -H II AND C algebr e rH t-H CNOO 1 1 1 m "2 GD ~ 05 T3 e ■W II s > e lO C5 r-H CO »C C3 CJ CO »C .-1 ,-ii-H tH CM COCO II 1 II is for Odd Harmonics as ank space opposite numbers, s -H II fiq e M * 2 3 Si 1? 1 1 1 t -H II c - «*■ 3*3 £2 88 SS 8 1 l II II 2 = a h O e -H II 0) ° £ 8 § o «4 e M » 3 Si & £8 * ■H H B l— I i— i CM CO i 1 -H II a c 32 g$£ & 2 i * -H II 3 a e II II II II 1 3 a a o o &. <M 1 KM U5 >£> II i— 1 i— 1 C3 1 + S» 5s> + 1 + 1 to (D N N CD «D O O 00 00 t- ti dodo S> 5» ^ o o * + ' 7 05 oq Ti(5 U5 I &3 © d "«5 o » s a Si 9> «.<* 328 APPENDIX Curve No. FORM E Separation of Odd and Even Harmonics for 36 Ordinate Analysis Vm' = \ (Vm — ym+u). J/m" = Hj/"» + Vm+is). Data Diff. Sum. TFt *m 9m 2»«' 2v m " 18 1 19 2 20 3 21 4 22 5 23 6 24 7 25 8 26 9 27 10 28 11 29 12 30 13 31 14 32 15 33 16 34 17 35 Use y m ' in analysis for odd harmonics. Carry y m " to Form H as data. Remarks. Computed by APPENDIX 329 k O ■ > P 05 o a a O M 8 OiHNeO't'CtONOOOSO'HNW^iOtOt II II II II II II II II 1 1 eQB ° QQQQ » ■. » v ~ Mil ^^ - ++IJ n i ii ii n n H ii 0Q 1 72 I OHN«*10»M»0 to s 1 1 t- CO U5 Tj< CO <N «■< O 1 ■ t 1 OHnM<nneNoo» e| « II 2 £ o O ON HAHHO ■v $ Am °8 s omtooNio O J9 £ T3 O -y ft, s3 03 9 ti til Is ii o oq u o 13 e BQ ■H tnt» i— 1 k, 1 .2 1 a; ■H | o o fa 1 ■5 c "a -H .-* KSt- rHCO r- || ■ 1 1 ^ c "*) ■H c II a '< s « c % ■H c 2 ^ iftt* t* II fa ~> ^ = -: M * 1! 5 1*14 in W5 7+ a § mm o'o + 1 coco CO «5 oooo do II II SSl a e + I ++ ■^ USlffl I 0**! II IMi a a a =3 II K It *■ i 1 1 + ii ii n 330 AIM'KNDIX fa I ODD o PS s o « a 5 O Z < Curve No. © HH ^ »« co 1 1 »-i 1— I 1— 1 II II 00 CN CO I i-H CO "5 CN ^ CO 00 > o fi ,jNm 'OcsGOt^-cO'OeC'— i ^oooooooo — ti :: — .-. — [- s ^^ ^^ ^^ ^3^ . ~ + I '+ I h-lt-H 00 (N I CO t-i U5 05 .-H CO lO tOOONOOOOCi »-CCOlOCOt^OOC5C5< OOOOOOOO^h 288§88ggg a$ >> £> II " oqoq oqcq II II 05*05 -HHII + 1 i I ' 1— I I— I APPENDIX 331 a w I £ W OS t> I— I u - o < - fc « w a c C £ a I ■ 1 1 o e 1 1 t 1 e OrHNCC^ Hl 0fflN0005 II II II II II II II II + + + + °5 5Q ^3 &2 &2 &2 MM ++++++ 1 1 II II II II II II H II 5 6 1 1 OhNC0t)<i0ON00 01 BD <J 1 NCOiO^MNhO 3 OS Q 1 1 t m 1 ©i-<NW<i<"5cOt>.00CS 05 + «ak a H + Curve No. 1 1 c "5! t~ a c. EL g § 3 O -3 E O s OMtOOINW >-H I-H 3 o 1 I a: to <£to to + + 1 1 CD CD CD CO CO «D CD CO 00 00 00 00 OOOO + 1 + 1 &ef Scf 6q tq „ I I + + « «5 "5 "3 lO + + + I I tq bq tq [33 it: 1 1 \+ + &3 fcq Sq fcq lag ssj s» »» a» s> II II II II II "3 ^ ftq cq tt - + + ^ ^ ^ fcq cq + + + I I ■^ ^ -^ cq cq II II II II II &CJ &3 &B <i to 332 APPENDIX Curve No. « s 1 o s to o ■ sis a § c I I I I OS »c i— i eo i-- I I I I II II ^ I* II II II II I I I I I i-iooo© '8°8m ^ 2+ I + 02 I I I I 05 oq H II 05 oq II II 69 o^cc ri g N S 3 MtOOOOl dodo ~ i- ■ -.c X II II 0505 + 1 + 1 i m ' — — University of Toronto DO NOT REMOVE THE CARD FROM THIS POCKET Acme Library Card Pocket Under Pat. "Ref. Index File" Made by LIBRARY BUREAU im.||||l|||pillllll W Kill i"ll 1! it fill 11 1 i I 'I' •*'•'" I'll •Pill j w illlj! ill ijii-'-ii'lilS 1